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PLANK AND SPHERICAL TRIGONOMETRY 



BOOKS BY 

LYMAN M. KELLS, WILLIS F. KERN, 
and JAMES R. BLAND 

PLANE AND SPHERICAL TRIGONOMETRY 
Second Edition 
6x9, Illustrated. 
With tables, 516 pages. 
Without tables, 401 pages. 

PLANE TRIGONOMETRY 
Second Edition 
6x9, Illustrated. 
With tables, 418 pages. 
Without tables, 303 pages. 

LOGARITHMIC AND TRIGONOMETRIC TABLES 
118 pages, 6x9. 



















Hipparchus (c. 140 B. c.) definitely began the science of trigonometry by working 
out a table of chords, that is, of double sines of half the angle. 

Claude Ptolemy (c. 150) did for astronomy what Euclid did for plane geometry. 
His work on astronomy was a standard of excellence for many centuries. 

John Napier (1550-1617) invented logarithms. This remarkable invention 
affects the whole world with constantly increasing power. 



Leonard Euler (1707-1783) was, in a sense, the creator of modern mathematical 
expression. The equation e ix =* cos x + i sin x is called by his name. 



PLANE AND SPHERICAL 
TEIGONOMETRT 



BY 

LYMAN M. KELLS, PH.D. 

Associate Professor of Mathematics 

WILLIS F. KERN 

Associate Professor of Mathematics 
AND 

JAMES R. BLAND 

Associate Professor of Mathematics 
All at the United States Naval Academy 



SECOND EDITION 
NINTH IMPRESSION 



McGRAW-HILL BOOK COMPANY, INC. 

NEW YORK AND LONDON 
1940 



COPYRIGHT, 1935, 1940, BY THE^V ' 
McGRAw-HiLi, BOOK COMPANY, INC. 



PRINTED IN THE UNITED STATES OF AMERICA 

All rights reserved. This book, or 

parts thereof, may not be reproduced 

in any form without permission of 

the publishers. 



THE MAPLE PRESS COMPANY, YORK, PA. 



PREFACE 

The improvements attempted in this revision fall roughly into 
three main categories, namely: those obtained by enlarging the 
old lists of problems and by supplying new lists; those obtained by 
employing a psychological approach to trigonometry and to each 
of its main branches; and those obtained by using freely sugges- 
tions and criticisms derived from classroom experience. 

Each original list of problems has been greatly amplified, and 
new review lists have been introduced. These are supplemented 
by numerous pictures which are interesting in themselves, and 
which serve the purpose of visually calling the student's attention 
to the direct nature of the applications. They suggest to his 
mind the actual situation and the reality of the problem. These 
problems and pictures will provide both teacher and student with 
a, wide range of motivating and interesting material. 

The greatest of care has been exercised in presenting an intro- 
ductory chapter that will at once grip the student's interest and 
give him a firm foundation for the trigonometrical superstructure. 
A number of elementary applications of fundamental ideas to 
familiar everyday situations illustrate both principle and applica- 
tion; they make the definitions appear natural and useful and thus 
furnish initial motivation. These lead to practical problems with 
figures and to exercises in which the right triangle appears in 
various positions and others in which it appears as part of a recti- 
linear figure. Solving these exercises teaches the student the 
practical value and power of trigonometry while giving that 
thorough working knowledge of the definitions which enables the 
student to grasp easily the deductions flowing from them. The 
same care has been used to follow closely the laws of learning in 
presenting each new phase of the subject. 

A number of the users of the text have given constructive 
criticisms of many special topics, and the treatment of various 
ideas has been discussed almost daily by the teachers of mathe- 



viii PREFACE 

matics at the Naval Academy. Criticisms and suggestions have 
been freely employed to make many minor improvements. 

The authors gladly take this opportunity to thank all those 
who have helped with constructive ideas. We are especially 
indebted to Commander W. P. 0. Clarke, who furnished us with 
many of our newest applications, and to Professor James B. Scar- 
borough, who read the manuscript completely. 

LYMAN M. KELLS, 
WILLIS F. KERN, 
JAMES R. BLAND. 
ANNAPOLIS, MD., 
July, 1940. 



CONTENTS 

PAGE 

PREFACE vii 

PLANE TRIGONOMETRY 

CHAPTER I 

TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 
ART. 

1. Introduction. . . 3 

2. Ratio . . . ... 4 

3. The Tangent, the Sine, and the Cosine 5 

4. The Cotangent, the Secant, and the Cosecant 9 

5. Trigonometric Functions of 45, 30, 00, 0, 90 . . . . 12 

6. Table of Values of Trigonometric Functions . 15 

7. Finding Heights and Distances by Means of Trigonometric Func- 

tions . .... ... 16 

8. Solving Rectilinear Figures 20 

9. Miscellaneous Exercises 23 

CHAPTER II 

FUNDAMENTAL RELATIONS AMONG THE TRIGONOMETRIC 

FUNCTIONS 

10. Introduction 28 

11. Simple Relations 28 

12. Identities and Conditional Equations . 30 

13. Relations Derived from the Pythagorean Theorem 32 

14. Verification of Identities ,34 

15. Formulas from Right Triangle 37 

16. Length of Line Segments 40 

17. Miscellaneous Exercises 44 

CHAPTER III 
GENERAL DEFINITIONS OF TRIGONOMETRIC FUNCTIONS 

18. Definition of Angle 48 

19. Rectangular Coordinates 49 

20. Definitions of the Trigonometric Functions of Any Angle .... 51 

21. Observations 55 

22. Values of Trigonometric Functions for Special Angles 55 

23. Fundamental Identities 58 

24. Expressing a Trigonometric Function of Any Angle as a Function 

of an Acute Angle . .58 

ix 



x CONTENTS 

ART. , PAGE 

25. Functions of and 180 6 in Terms of Functions of . . . 61 

26. Miscellaneous Exercises 64 

CHAPTER IV 
RIGHT TRIANGLES 

27. Introduction 67 

28. Accuracy 67 

29. Tables of Natural Trigonometric Functions 68 

30. Solving Right Triangles 70 

31. Definitions 72 

32. Solution of the Right Triangle by Slide Rule 76 

33. Slide-rule Solution of a Right Triangle When Two Legs Are Known 78 

34. Table of Logarithms of Trigonometric Functions . . . 79 

35. To Find the Logarithms of a Trigonometric Function of an Anglo 79 

36. To Find the Angle When the Logarithm Is Given 81 

37. Solution of the Right Triangle by Means of Logarithms .... 82 

38. Solution of Rectilinear Figures ... 85 

39. Miscellaneous Exercises 89 

CHAPTER V 
FORMULAS AND GRAPHS 

40. Introduction 94 

41. The Radian 94 

42. Length of Circular Arc 96 

43. Functions of 90 - 6 99 

44. Functions of 90 + 0, 270 + 0, 180 0, -0 101 

45. Functions of (k 90 0) 104 

46. Graph of y = sin x 106 

47. Graph of y cos x 108 

48. Graph of y = tan x 109 

49. Graphs of y cot x, y sec x, y = esc x Ill 

50. Graphs and Periods of the Trigonometric Functions of k0 . . . . 113 

51. Miscellaneous Exercises 116 

CHAPTER VI 
GENERAL FORMULAS 

52. The Addition Formulas 120 

53. Proof of the Addition Formulas. Special Case 120 

54. Removal of Restrictions on the Addition Formulas 124 

55. Addition and Subtraction Formulas for the Tangent 125 

56. The Double-angle Formulas and the Half-angle Formulas. . . . 128 

57. Conversion Formulas 133 

58. Miscellaneous Exercises 136 

CHAPTER VII 
IMPORTANT FORMULAS RELATING TO TRIANGLES 

59. Law of Sines 140 

60. Law of Tangents. Mollweide's Equations 144 



CONTENTS xi 

ART. P AQB 

61. Law of Cosines 146 

62. Miscellaneous Exercises 149 

CHAPTER VIII 
OBLIQUE TRIANGLES 

63. Introduction 152 

64. Form for Computation by Logarithms to Be Used in the Solution 

of Oblique Triangles 153 

65. Case I. Given One Side and Two Angles. ... .... 153 

66. Case II. Given Two Sides and the Angle Opposite One of Them 156 

67. Case III. Given Two Sides and the Included Angle 161 

68. The Half-angle Formulas 164 

69. Case IV. Given Three Sides 167 

70. Summary 169 

71. Miscellaneous . . 170 

CHAPTER IX 
INVEHSK TRIGONOMETRIC FUNCTIONS 

72. Inverse Trigonometric Functions . 177 

73. Graphs of the Inverse Trigonometric Functions . 178 

74. Representation of the General Value of the Inverse Trigonometric 

Functions 180 

75. Principal Values . 182 

76. Relations among the Inverse Functions ... ... 184 

77. Examples Involving Inverse Trigonometric Functions 185 

78. Trigonometric Equations 189 

79. Special Types of Trigonometric Equation , . . . 192 

80. Equations Involving Inverse Functions . 195 

81. Miscellaneous Exercises 197 

CHAPTER X 
COMPLEX NUMBERS 

82. Pure Imaginary Numbers 200 

83. Complex Numbers 201 

84. Operations Involving Complex Numbers ... . . 201 

85. Geometrical Representation of Complex Numbers 202 

86. Polar Form of a Complex Number 203 

87. Multiplication of Complex Numbers in Polar Form 205 

88. The Quotient of Two Complex Numbers in Polar Form . . 206 

89. Powers and Roots of Complex Numbers, De Moivre's Theorem . 208 

90. Exponential Forms of a Complex Number 211 

91. The Hyperbolic Functions. . 212 

92. Miscellaneous Exercises. . . 214 

CHAPTER XI 
LOGARITHMS 

93. Introduction 216 

94. Laws of Exponents 216 



xii CONTENTS 

ART. 

95. Definition of a Logarithm 217 

96. Laws of Logarithms 218 

97. Common Logarithms. Characteristic . . . 221 

98. Effect of Changing the Decimal Point in a Number 223 

99. The Mantissa .... 224 

100. To Find the Logarithm of a Number ... . 224 

101. Interpolation . 225 

102. To Find the Number Corresponding to a Given Logarithm . 226 

103. The Use of Logarithms in Computations . . 227 

104. Cologarithms 229 

105. Computation by Logarithms . . . 230 

106. Suggestions on Computing by Logarithms. . . 231 

107. Change of Base in Logarithms . . ... 235 

108. Solution of Equations of the Form x = a 6 , a = d> . . 235 

109. Graph of y = logio x 237 

110. Miscellaneous Exercises. . . . . 238 

CHAPTER XII 
THE SLIDE RULE 

111. Introduction . . . .... 241 

112. Reading the Scales . . 241 

113. Accuracy of the Slide Rule .... 243 

114. Definitions ... 243 

115. Multiplication . . . 244 

1 16. Either Index May Be Used . . . . ... 245 

117. Division 246 

118. Use o Scales Wand CF (Folded Scales) .... 247 

119. The Proportion Principle ... . . . . 249 

120. Use of the CI Scale 250 

121. Combined Multiplication and Division 251 

122. Square Roots . ... ... 253 

123. Combined Operations Involving Square Roots 254 

124. The S (Sine) and ST (Sine Tangent) Scales 256 

125. The T (Tangent) Scale 257 

126. Combined Operations . . . 258 

127. Solving a Triangle by Means of the Law of Sines 259 

128. To Solve a Right Triangle When Two Legs Are Given . 262 

129. To Solve a Triangle in Which Two Sides and the Included Angle 

Are Given . . 263 

130. To Solve a Triangle in Which Three Sides Are Given . 264 

131. To Change Radians to Degrees or Degrees to Radians . . . 265 

SPHERICAL TRIGONOMETRY 

CHAPTER XIII 
THE RIGHT SPHERICAL TRIANGLE 

132. Introduction 269 

133. The Spherical Triangle 269 



CONTENTS xiii 

ART. PAGE 

134. Formulas Relating to the Right Spherical Triangle 271 

135. Napier's Rules 274 

136. Two Important Rules. . . 277 

137. Solution of Right Spherical Triangles 278 

138. The Ambiguous Case 281 

139. Polar Triangles 282 

140. Quadrantal Triangles 284 

141. Miscellaneous 285 

CHAPTER XIV 
THE OBLIQUK SPHERICAL TRIANGLE 

142. Law of Sines 289 

143. Law of Cosines for Sides 291 

144. Law of Cosines for Angles 293 

145. The Six Cases 295 

146. The Half-angle Formulas . . 295 

147. Cases I and II. Given Throo Sides or Given Three Angles . . . 298 

148. Napier's Analogies ... 300 

149. Cases III and IV. Given Two Sides and the Included Angle or 

Given Two Angles and the Included Side 303 

150 Cases V and VI. Two of the Given Parts Are Opposite. Double 

Solutions 304 

151. Miscellaneous Exercises 307 

CHAPTER XV 

VARIOUS METHODS OF SOLVING OBLIQUE SPHERICAL 

TRIANGLES 

152. Introduction. . . . 309 

153. Cases III and IV. 309 

154. Observations and Illustrative Example 310 

155. Case III. Alternate Method ... 312 

156. Ilaversine Solution of Case III 315 

157. Cases V and VI 316 

158. Observations and Illustrative Example 318 

159. Cases I and II 319 

160. Miscellaneous Exercises. . . 320 

CHAPTER XVI 
APPLICATIONS 

161. Nature of Applications 322 

162. Definitions and Notations 322 

163. Course and Distance 323 

164. The Celestial Sphere 328 

165. The Astronomical Triangle 330 

166. Given t, d, L; to Find h and Z 331 

167. To Find the Time and Amplitude of Sunrise 333 



xiv CONTENTS 

ART. PAOB 

168. To Find the Time of Day 335 

169. Ecliptic. Equinoxes. Right Ascension. Sidereal Time .... 337 

170. The Time Sight 340 

171. Meridian Altitude. To Find the Latitude of a Place on the Earth 341 

172. Given t, d, h, to Find L 343 

173. Miscellaneous Exercises 344 

APPENDIX 351 

INDEX 365 

ANSWERS 369 



GREEK ALPHABET 

Letters Names Letters Names Letters Names 

a Alpha t lota p Rho 

ft Beta K Kappa <r $ Sigma 

7 Gamma X Lambda T Tau 

6 Delta n Mu u. ; Upsilon 

e Epsilon v Nu ^> Phi 

f Zeta Xi x Chi 

T/ Eta o Omicron \l/ Psi 

Theta TT Pi o> Omega 



LIST OF SYMBOLS 

= , read is identical with. 

?^, read is not equal to. 

< , read is less than. 

>, read is greater than. 

^, read is less than or equal to. 

;> , read is greater than or equal to. 

(x, y), read point whose coordinates arc x cmd y. 



PLANE TRIGONOMETRY 



CHAPTER I 

TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 

1. Introduction. A cadet who was 6 ft. tall found that his 
shadow was 3 ft. long (sec Fig. 1). He argued that since his 
height was twice the length of his shadow, the height of a near-by 
flagpole must be twice the length of its shadow. He then 
measured the shadow of the flagpole and found that it was 7 ft. 
long. He concluded that the height of the flagpole was twice the 




FICJ. 1. 

length of its shadow, or 2 X 7 ft. = 14 ft. In other words, by 
observing that the ratio of the height of a certain right triangle 
to its base was y, he found the height of a flagpole without 
measuring it. 

This is a very elementary illustration of what navigators, 
surveyors, engineers, and others do with trigonometry. By 
applying the complete theory of the ratios of the sides of a right 
triangle (that is, trigonometry) to data obtained by measure- 
ments, they find inaccessible heights of mountains and distances 
through them; distances across lakes, rivers, and inaccessible 
swamps; boundaries of fields and countries; and positions at sea. 
Engineers use trigonometry every day in their work of con- 
structing large buildings, bridges, and roads; astronomers use it to 

3 



4 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 

determine the time by which clocks are regulated ; surveyors use 
it constantly to find all sorts of heights, distances, and directions; 
and navigators use it to compute latitude, longitude, and course 
at sea. 

Trigonometry has other very important uses. The ratios of 
the sides of right triangles are capable of describing phenomena 
of a periodic nature such as the to-and-f ro motion (j. a pendulum 
and the motion of waves. Consequently, they play an important 
part in the theory of light and sound, in electrical theory, in wave 
analysis, and in all investigations dealing with phenomena of a 
vibratory character. Hence, although most of the problems 
stated in this book to illustrate practical phases of trigonometry 
deal with heights of inaccessible objects and distances, a largo 
number of exercises will help to familiarize the student with a 
class of functions of great importance in more advanced mathe- 
matical theory. 

2. Ratio. At the very base of trigonometry lies the idea of 
ratio. The ratio of a number a to a number 6 is the quotient 

a divided by 6, that is, a/6; 
>C' the ratio of two line segments 
is the ratio of the length of 
one segment to the length of 
the other and is independent 

of the unit of mea sure; the 
ratio of a line segment 1 mile 
long to another 2 miles long is ^, whether the lengths be 
expressed in miles or in feet. 

One of the main reasons for the usefulness of trigonometry is 
that it furnishes a method of finding ratios associated with 
angles. One gets some idea of the importance of a knowledge 
of these ratios by considering the usefulness of models of mach-nes, 
of blueprints of buildings, and of various kinds of ma^s The 
plane angle made by two straight lines in the model is th * same as 
the angle made by the corresponding lines in the actual ^tnurcure; 
therefore the ratios associated with the angles in the mDdel will 
be the same as those in the corresponding angles in the structure 
represented. Thus the angles made by corresponding lines m the 
similar diamonds represented in Fig. 2 are equal. The cadet 
mentioned in 1 found the height of the flagpole by using the ratio 




3] THE TANGENT, SINE, AND COSINE 5 

of the length of an object to that of its shadow. A traveler can 
find distances approximately by using the fact that map dis- 
tances have the same ratio as actual distances. 

Three important ratios, the fundamental quantities of trigo- 
nometry, will be considered in the next article. If A represents 
any angle, the throe ratios are caUed 'the tangent of A, the sine 
of A, and the cosine of A, respectively. 

^ "The tangent, the sine, and the cosine. If every value of a 
variable x within a certain interval is associated with a value of 
another variably in such a Way that when x is given y is deter- 
mined, then y idHt,>f unction of x., Thus the area of a square is a 
function of its skje, since when the side is given the area is deter- 
mined; the distance flared by a car running at a constant speed 
is a function of the timcjl,; Later we shall find that certain ratios 
of lengths of line segments Ere func- 
tion^ of a iv 

Conside^Hfr acute angle such as 
a'ngle ^jof ^K3. "From any point B 
on one side or the angle drop a per- 
pendicSar to the other side, meeting 
it in C, and consider the ratio CB/AC. FlG ' 3 - 

The question arises: Is the value of this ratio determined when the 
angle is given? The following argument shows that it is. Let 
Bid represent any other line drawn from a point B\ on one side 
of the angle perpendicular to the other side and meeting it in d. 
Then the triangles ABC and AB\C^ are similar since they are 
right triangles having an acute angle of one equal to an acute 
angle of the other. Since corresponding sides of similar triangles 
are proportional, we have 




AC ~ Ad* 

Thus the value of the ratio CB/AC is determined when an acute 
angle is given. Consequently, in accordance with the definition 
just given, this ratio is a function of the acute angle. The ratio 
CB/AC in Fig. 3 is named the tangent of angle A, and we write 

rin 

tan A = . (2) 



6 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 



Also, two acute angles that have the same tangent are equal. 
Let A and A' in Fig. 4 be two angles such that 

tan A = tan A'. (3) 

Construct the right triangles shown in Fig. 4. Then, from (3) 

B' 

>^ 

B 




90% 



J C < 



FIG. 4. 



and the definition (2), 
CB 



= tan A = tan A 1 = 






(4) 



Hence the two triangles in Fig. 2 are similar, hlKig an angle 
(90) of one equal to an angle of the other and the including sides 
proportional. Therefore angle A and angle A', being correspond- 
ing angles of similar triangles, are equal. 

For convenience, we shall indicate that an angle is a right 

angle by drawing a small square at 
its vertex. Thus the small square 
at C in Fig. 5 shows that angle C is a 
right angle. 

Two other ratios, besides the tan- 
gent of an angle, are very important. 
The ratio CB/AB in Fig. 5 is called 
the sine of angle A, and the ratio 
AC/AB is called the cosine of angle A. Using the abbreviations 
cos for cosine and sin for sine, we have from Fig. 5 




Adjacent leg 
Fio. 5. 



sin A - PP site lg g, 

Dill f\ ^ ~ I 

hypotenuse 

A adjacent leg 
cos A = !-= T i 
hypotenuse 



tan A = 



adjacent leg 



(6) 



3] 



THE TANGENT, SINE, AND COSINE 



These ratios are called trigonometric functions. By using the 
same line of reasoning applied in the case of the tangent, we 
can show that the value of each of the three trigonometric functions 
of an acute angle is determined when the acute angle is given. 
Furthermore, it can be shown that if the value of any one of the three 
trigonometric functions of an acute angle is equal to the value of 
the same function of a second acute angle, the two acute angles are 
equal. 

Example 1. Find the values of the three trigonometric func- 
tions of an angle A if its sine is 5. 

Solution. Draw a right triangle having its hypotenuse 5 units 
long and one leg 3 units long (see Fig. 6). The acute angle 
opposite the 3-unit leg is angle A , since its sine 
is f . Also, the side AC = \/25 9 = 4. 
Then, from Fig. 6, we read in accordance 
with the definitions (5) 

sin A = f , 
cos A = %, 
tan A = f . 



B 



4 
FIG. 6. 



Example 2. A surveyor wishing to find the height of a light- 
house measures the angle A at a 
point 120 ft. from its base. His 
findings are represented in Fig. 7, 
where tan A = f . What is the 
height of the lighthouse? 

Solution. From triangle ABC we 
read 



, A CB 
tan A = 



or tan A 



CB 
120' 




Solving this equation for CB and replacing tan A by its value f-, 
we obtain 

CB = 120 tan A = 120(f) = 80 ft* 



* Throughout this book the answers to illustrative examples will be printed 
in boldface characters. 



8 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 



EXERCISES 

1. From each of the Figs. 8, 9, 10, 11, 12, and 13 read tan A and 
tan B. 

A 



Vl3 




B 





FIG. 11. FIG. 12. 

2. From each of Figs. 14, 15, 10, and 17 obtain sin A, cos A, and 
tan A. 





4 
FIQ. 14. FIG. 15. FIG. 16. 

3. If sin A = TC, find cos A and tan A. 

4. If cos A -fat find sin A and tan A. 

5. If tan A = ^, find sin A and cos A. 

6. If sin A = -pr, find cos A and tan A. 

7. If cos A = !|-, find sin A and tan A. 

8. If cos A = ff, find sin A and tan A. 

9. If sin A = ;= show that sin A = cos A. 

V2 

10. For angle A in Fig. 14, show that 



10 
Fio. 17. 



(a) sin A cos A = 



(c) (shTA) 2 + (cosA) 2 = 1, 
1 



4] 



THE COTANGENT, SECANT, AND COSECANT 




11. An observer at A (see Fig. 18), 
1110 ft. from and on a level with the 
base of the Washington Monument, 
sights its top and finds that the angle ^ 
A is such that tan A = -%. Find the ^*-- 
hcight of the monument. 

12. A base line AC 350 ft. in length is laid along one bank of a river. 
On the opposite bank a point B is located so that CB is perpendicular to 
AC. The tangent of the angle CAB is then measured and found to be 
-&. Find the width of the river. 



1110' 
FIG. 18. 



13. Figure 10 represents a ladder leaning 
against the side of a house. If the ladder is 36 
ft. long and cos A = -J-, how far is the foot of 
the ladder from the house? 




FIG. 19. 



14. The length of string between a kite and a point on the ground is 
225 ft. If the string is straight and makes with the level ground an 
angle whose tangent is nr, how high is the kite? 

Jk 

fl 

ire r_v_i -=^0 

C 

FIG. 20. 

aa Figure 20 shows the relative positions of a point and two oil 
wells, A and C, 300 ft. apart. An observer at finds that the sine of 
angle AOC is J-. What is his distance from the well at A? 

4. The cotangent, the secant, and the cosecant. Besides the 
three ratios (5) of pairs of sides of a right triangle, there are three 
others got by writing the reciprocals of the ratios in (5). The 
reciprocals of tan A, cos A, and sin A are called, respectively, 
cotangent A, secant A, and cosecant A, and are represented by 
cot A, sec Ay and esc A. 



10 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 



Referring to the right triangle in Fig. 21, we make the following 
definitions: 




Adjacent leg 
FIG. 21. 



. . adjacent leg 

cot A = ., , 

opposite leg 

hypotenuse 
adjacent leg' 
- _ hypotenuse 
~~ opposite leg" 



sec A = 



esc 



(6) 



6 

FKJ. 22. 



Just as before*, the value of each trigonometric function is 
B determined when the acute angle is given ; 
and if the value of any one of the six trig- 
onometric functions of an acute angle is 
equal to the value of the same function 
of a second acute* angle, the* two acute 
angles are* equal. 

Since y/x = 1 -f- (x/y), it appears from 
the definitions (5) and (6) and Fig. 22 that 



(7) 



It will be well for the student to think of esc A, sec A, and 
cot A as reciprocals of sin A, cos A, and tan A, respectively; 
thus, to find esc A, think of the fraction for sin A and then write 
its reciprocal. 

Use is sometimes made of the trigonometric functions defined 
as follows: 



1 

CSC A 

sec A 
cot A 


a 
c 


1 

a/c 
1 


1 A 

sin A 1 
1 


b 
b 


b/e - 
1 


cos A' 

1 


a 


a/b 


tan A i 



versed sine of (written vers 0) = 1 cos 6, 

haversine of B (written hav 0) = ^(1 cos 0),] 

coversed sine of (written covers 0) = 1 sin B. 

EXERCISES 



(8) 



1. In each of the Figs. 23, 24, 25, 26, 27, and 28 write the six trigono- 
metric functions of angle A. 



4] 



THE COTANGENT, SECANT, AND COSECANT 



11 






Fin. 24. 



5 
FIG. 25. 






FIG. 27. 

2. The sides of a right triangle are 5, 12, and 13, respectively. Read 
the values of the trigonometric functions of the angle opposite the 
5-unit leg. Also read the functions of the angle opposite the 12-unit leg. 

3. Find the values of all the trigonometric functions of an acute 
angle having (a) its sine equal to f ; (6) its tangent equal to ^; (c) its 
cosine equal to 1. 

4. If sin A T, find the value of 



(a) (sin A) 2 + (cos 



(6) (esc AY - (cot AY. 

cot i 



6. Given that sin D = f , tan E = ^, cos F = 
show that the following equations are true: 

(a) (cos 7)) 2 sec G cos E ^. 

(6) (esc D) 2 cot F cot # = 2. 

(c) sec E tan F cot (r sin G tan Z) = V. 

(d) sin D esc E sec G cos # = 2. 

(e) esc D cot F esc G cos E = 



6. The relative positions of the point A at the bow of a ship 300 ft. 
long, C at its stern, and B on a near-by submarine are shown in Fig. 29. 



12 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 

If the tangent of angle ABC is f and angle ACB is 90, about how far is 
the submarine from the ship? 




FIG. 29. 



7. The^central pole of a circular tent is 30 ft. high and is fastened at 
the top by ropes to stakes set in the ground. Each rope makes an angle 
A with the ground such that esc A = f . Find the length of each rope. 




B 220' C 
FIG. 30. 



8. Figure 30 represents a radio tower. AC is 
a cable anchored at point C on a level with the 
base of the tower. The angle C made by the 
cable with the horizontal is such that sec C = f . 
If the distance from C to the center B of the base is 
220 ft., find the length of the cable. 



5. Trigonometric functions of 46, 30, 60, 0, 90. If a 

square be constructed with sides 1 unit in length, its diagonal 
will be -y/1 2 + I 2 = \/2 units long and will make a 45 angle 
w with a side (see Fig. 31). Then, from the 
B ' triangle ABC (Fig. 31), we read in accord- 
ance with definitions (5) and (6) 

sin 45 = 1/V2 = 0.7071, 
cos 45 = l/\/2 => 0.7071, 
tan 45 = 1/1 = 1.0000, 
esc 45 = -v/2/1 = 1.4142, 

sec 45 = \/2/l = 1.4142, 
cot 45 = 1/1 = 1.0000. 




Fio. 31. 



TRIGONOMETRIC FUNCTIONS OF 45, 30, 60, 0, 90 13 



If an equilateral triangle be constructed with sides 2 units in 
length and if the bisector of one of its angles be drawn, this 
bisector will have a length of \/3 units, will make a 30 angle 
with each of two sides, and will be perpendicular to the third side 
(see Fig. 32). Hence, from the triangle ABC 
of Fig. 32, we read 



' sin 30 = 1/2 = 0.5000, 
cos 30 = A/3/2 = 0.8660, 
tan 30 = l/A/3 = Q.5774, 
esc 30 = 2/1 = 2.0000, 
sec 30 9 = 2/\/3 = L1547, 
cot 30 = V3/1 = 1.7321. 




FIG. 32. 




Placing the triangle of Fig. 32 in the position shown in Fig. 33, 
we read from triangle ABC 

sin 60 = \/3/2 = 0.8660, 
cos 60 = 1/2 = 0.5000, 
tan 60 = V3/1 = 1.7321, 
esc 60 = 2/V3 = 1.1547, 
sec 60 = 2/1 = 2.0000, 
cot 60 = l/A/3 = 0.5774. 

The trigonometric functions of are, 
by definition, the results obtained by 
placing opposite log equal to zero and adjacent leg equal 
to the hypotenuse in the definitions (5) and (6). Hence 
they may be read from Fig. 34. ^ B 

Since BC = and since divi- ^ ? f . .no 

sion by zero is excluded from l 90 C 

algebraic operations, it appears IQ * ' 

that esc and cot are undefined. Nevertheless, we write esc 
= oo , cot = QO , and mean by these symbols that, as an acute 
angle varies and approaches zero as a limit, the values of esc 
and cot vary and become greater and greater without limit. 
Hence, from Fig. 34, we write 



sin = 0, 
cos = 1, 
tan = 0, 



CSC = 00 , 

sec = 1, 
cot = oo . 



14 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP.! 
B 



Similarly, from Fig. 35, we write 

sin 90 = 1, esc 90 = 1, 

cos 90 = 0, sec 90 = oo , 

tan 90 = oo , cot 90 = 0. 



EXERCISES 

1. Draw a right triangle, one of whose acute angles is 30. Assign 
appropriate lengths to the sides of this right triangle, and from it read 
the values of the trigonometric functions of 30 and of 60. 

2. Find approximately the values of the trigonometric functions of 1' 
by reading them from Fig. 36. From this same figure read the approxi- 
mate values of tEe trigonometric functions of 8959'. 



30.000291 




3. From Fig. 34 read the values of the trigonometric functions of 
and of 90. 

4. Draw a triangle from which may be read the values of the trigono- 
metric functions of an angle A whose sine is -rr. From this figure read 
the values of the trigonometric functions of A and of 90 A. 

5. If sec A = 2, write the trigonometric functions of A. 

6. If tan A = 1, write the trigonometric functions of A. 

7. Prove that cos 60 = 2 cos 2 30 - 1. 

sec 60 

8. Prove that tan 30 = -, ~ , 1X ^^o- 

(sec 60 + 1) esc 60 

9. Find the values of each of the following: 

(a) tan 30 sin 60 sec 30 cot 45. 

($) esc 45 sin 90 tan 60 cos 0. 

(c) cos 45 esc 45 - tan 45 tan 0. 
y) n 30 gin 45 cos esc 60 cot 60. 



6] TABLE OF VALUES OF TRIGONOMETRIC FUNCTIONS 15 

10. Show that 

(a) sin 90 = sin 30 cos 60 + cos 30 sin 60. 
(6) cos 30 = cos 60 cos 30 + sin 60 sin 30. 
(c) sin 30 = sin 60 cos 30 - cos 60 sin 30. 

11. If tan A = tari 45 cos 30 tan 60, find the trigonometric func- 
tions of A. 

12. That the formulas 

- sin (A + B) = sin A cos B + cos A sin B 
cos (A B) = cos A cos B + sin A sin B 

are true for all values of A and B will be proved in Chap. VI. In these 
formulas substitute A = 45, B 30, and evaluate the resulting right- 
hand members to obtain sin 75 and cos 15, respectively. . 

13. A tree stands at a certain distance from a straight road on which 
two milestones are located. The tree was observed from each mile- 
stone, and the angles between the lines of sight and the road were found 
to be 30 and 90, respectively. Find the distance from the tree to the 
road. 



14. The ladder leaning against the wall in Fig. 
37 is 45 ft. long. If it makes an angle of 60 with 
the horizontal, how far is the foot of the ladder 
from the wall? 




FIG. 37. 



116. jA farmer wishes to fence a field in the form of a right triangle. 
If GTO angle of the triangle is 45 and the hypotenuse is 200 yd., find the 
amount of fencing needed. 

6. Table of values of trigonometric functions. Approximate 
values of the trigonometric functions of certain angles have been 
computed and arranged in tabular form. The small table 
printed here gives, accurate to three decimal places, the values of 
six trigonometric functions for each of the angles 0, 5, 10, 
. . . , 90. 

The value of a desired function of an angle is found in the 

column headed by the name of the function and in the row 

aving as its first entry the number of degrees in the angle. For 



16 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 



example, in the column headed tan (tangent) and in the row 
having 25 as its first entry, read tan 25 = 0.466. 

Table of Trigonometric Functions 



Degrees 


sin 


cos 


tan 


cot 


sec 


CSC 





000 


1.000 


000 


00 


1.000 


oo 


5 


087 


0.996"" 


0.087 


11.430 


1.004 


11.474 


10 


0.174 


985 


176 


5 671 


1 015 


5 759 


15 


0.259 


966 


268 


3 732 


1 035^ 


3 864 


20 


0.342 


940 


364 


2.747 


1 064 


2 924 


25 


423 


906 


466 


2 145 


1 103 


2 366 


30 


500 


866 


577 


1 732 


1.155 


2 000 


35 


574"" 


819 


700 


1.428 


1 221 


1 743 


40 


643 


766 


839^ 


1 192 


1.305 


1 556 


45 


0.707 X 


707 


1.000 


' 1 000 


1 414 


1 414 


50 


766 


643 


1.192 


839 


1.556 


1 305 


55 


819 


574 


1 428 


700 


1 743 


1.221 


60 


0.866 


500 


1 732 


577 


2 000 


1 155 


65 


906 


0.423 


2 145 


466" 


2 366 


1 103 


70 


940 


0.342^ 


2 747 


364 


2 924 


I 064 


75 


0.966 


259 


3 732 


268 


3.864 


1 035" 


80 


0.985 


174 


5 671 


176 


5 759 


1 015 


85 


996 


087^ 


11 430 


087 


1 1 . 474 


1.004 


90 


1.000 


000 


CO 


000 


00 


1.000 



EXERCISES 

1. Use the table of this article to verify the following equations: 



(a) sin 35 = 0.574. 

(6) cos 70 = 0.342. 

(c) tan 40 = 0.839. 

(d) sec*15 = 1.035. 

(e) esc 75 = 1.035. 



(/) cot 65 = 0.466. 
(g) sin 45 = 0.707. 
(h) cos 85 = 0.087. 
(i) tan 85 = 11.430. 
(j) cos 5 = 0.996. 



2. Compute, accurate to three decimal places, sin 45, tan 45, 
sin 30, sec 30, esc 30, sin 60, sec 45, and compare with the values 
of these functions found from the table. 

7. Finding heights and distances by means of trigonometric 
functions. To find an unknown height or distance, one generally 
draws a figure representing the situation and then finds the part 



7] FINDING HEIGHTS AND DISTANCES 17 

of it corresponding to the unknown distance. The method of 
this article for finding the parts of a right triangle differs from the 
method used in preceding articles only in the way of getting the 
desired value of a trigonometric function; in preceding problems 
the function was given; here it must be found in the table of 
6. The following rule will be helpful at first. 

Rule. To find an unknown part of a right triangle when a side 
and another part are given: 

(a) Draw a figure on which are written the values of the known 
parts and a letter for the unknown part. 

(b) Read from the figure a formula connecting the known parts 
and the unknown part. 

(c) Replace any trigonometric function of a known angle in the 
result from step (b) by its value from the table of 6. 

(d) Solve the result from step (c) for the unknown part. 
The following example will illustrate the method. 

Example. An angle of a right triangle is 55, and the adjacent 
leg is 58 units. Find the remaining parts. 

Solution. In Fig. 38 the known parts of 
the right triangle are shown, and the letters 
B, a, c represent the unknown parts. Evi- 
dently B = 90 - 55 = 36. From the fig- 
ure read 

^ = tan 55. (a) 

y \6B ( 




From the table in 6, tan 55 = 1.428. ^ 

Substitute this value in (a), and solve the FIG. 38. 

result for a to obtain 

a = 58(1.428) = 82.8. 

Repeat the procedure to find c. From Fig. 38, 


TQ = sec 55. . (b) 

Oo 

Replace sec 55 by 1.743, its value from the table of 6, in (b), 
and solve the result for c to obtain 

c = 58(1.743) = 101.1. 



18 TRIGONOMETRIC 'Fl'NVTlONSOF AN ACL'TK ANGLE [CHAP. I 



A number is rounded off to three significant figures when it is 
expressed as nearly as possible by means of a first digit different 
from zero, two digits immediately following the first, and enough 
zeros to place the decimal point. Thus the figures 84321, 
0.05436, 0.5985, 0.5996, when rounded off to three significant 
figures, become 84300, 0.0544, 0.598, 0.600, respectively. 

In order to avoid indicating more accuracy than is warranted 
when a table accurate to three decimal places is used, round all 
answers off to three significant figures unless the first significant 
digit is 1 ; in this latter case round the answer to four significant 
figures. 

EXERCISES 

1. Find the unknown parts of the triangles of Figs. 39 to 42: 



B 





\50 6 



75 



FIG. 39. 



FIG. 40. 



800 
FIG. 41. 




2. In each of the following exercises, c refers to the hypotenuse of 
a right triangle, a to the leg opposite the acute angle .4, and b to the 
leg opposite the acute angle B. Solve each of the right triangles in 
which the known parts are, 



(a) c = 85, 
A = 35. 

(6) a = 200, 
B = 80. 

(c) a = 500, 
A = 55. 



(d) B = 75, 

c - 20. 

(e) c = 100, 
A = 25. 

(/) b = 60, 
B = 70. 



3. The hypotenuse of a right triangle is 800 ft., and sin A = [. 
Find the legs of the triangle. 

4. The following data refer to right triangles. In each case find the 
unknown sides. 



(a) c = 520, sin A = f . 

(b) a = 880, cos A = A 

(c) 6 = 34, tan B = |. 



(d) c = 250, cot B - 

(e) a = 173, esc B = 3. 
(/) b = 284, sin B = i 



FINDING HEIGHTS AND DISTANCES 



19 



5. A surveyor wishing to find the 
height of a tower, represented by MN 
in Fig. 43, stands 90 ft. from its base, 
measures the angle A, and finds it to 
be 3,5. If the surveyor's eye is 5 ft. 
above the ground, find the height of 
the tower. 




N 



FICJ. 43. 



6. A city block is in the form of a right triangle with a hypotenuse 
of 300 ft. If one angle is 35, find the lengths of the other two sides. 



7. In order to find the distance 
from T to an inaccessible point A 
(see Fig. 44), line Cli, 100 ft. long, 
was laid off perpendicular to (M, and 
angle ('HA was found to be 70. 
Find the distance ('A. 




Fiu. 44 



8. At a point .5.5 ft. from the base of a flagpole thai is standing on 
level ground the angle of elevation of the top flf the pole is 50. Find 
the height of the flagpole, correct to the nearest foot. 



9. A guy wire from a point 5 ft. 
from the top of a telephone pole makes 
Jin angle of (io \vith the level ground 
and is anchored 1.5 ft. from the base of 
the pole, us shown in Fig. 4;">. How 
high is the pole? 




FIG. 45. 



tfft An airplane starts from a station and rises at an angle of 10 with 
the horizontal. By how many feet will it clear a vertical wall 100 ft. 
high and 900 ft. from the station? 

\A An observer in a captive balloon is 985 yd. above level ground. 
The line of direction of the enemy's outpost makes an angle of 80 with 
the vertical. How far away is the outpost? 



20 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 




150' 
FIG. 46. 



12. When the direction of the sun makes 
an angle of 35 with the horizontal, an oil 
derrick casts a shadow 150 ft. long. How 
high is the derrick (see Fig. 46)? 



13. In a certain quartz crystal two of the plane faces of the crystal 
meet at an angle of 50. If the perpendicular distance from a point A 
in one face to the other face is 3 cm., find the distance of A from the 
intersection of the t\vo faces. 

14. A plot of ground is in the form of a right triangle, with one leg 
10 yd. long and its adjacent angle 20. Find the length of a fence sur- 
rounding the plot. 

15. An observer in the airplane shown in Fig. 47 measures the angle 
ABC and finds it to be 35. He reads from his altimeter the altitude 
BC to be 3467 ft. What is the width AC of the island? 




FIG. 47. 

16. The vshortest side of a field in the form of a right triangle is 300 
ft. long. If the angle opposite this side is 40, find the area of the field. 

8. Solving rectilinear figures. If all lines in a figure are 
straight, the figure is said to be rectilinear. By applying 
repeatedly the method of solving right triangles explained in 7, 
all parts of a rectilinear figure can often be found in terms of given 



*] 



SOLVING RECTILINEAR FIGURES 



21 



parts. In simple cases, the method consists in locating a right 
triangle that can be solved and solving it, then finding the parts 
of a second right triangle that can be solved after the parts of the 
first one are obtained, then solving a third right triangle, etc. 
The following example will illustrate the method. 

Example. In Fig. 48, OD = 35 units, AB = 29 units, 
esc x = ^, tan y = J ^-. Find the 
lengths of all line segments. & 

Solution. Since esc x = -f-, Fig. 49 
may be used to find any function of 
x; similarly, Fig. 50 may be used to 
find any function of y. From triangle 
ODA, sin x = a/35; and from Fig. 49, 
sin x = 4. Therefore 



35 



or 



a = 28. 



Also from triangle ODA, cos x = 6/35, 
and from Fig. 49, cos x = f . 
Therefore 

b 3 
35 = * r 




6 = 21. 



Applying the Pythagorean theorem to triangle AOB, if b = 21, 
we have 

c 2 + 21 2 = 29 2 , or c = \/29 2 - 21 2 = 20. 

From triangle BOG, tan y = 
d/c = d/20, and, from Fig. 50, 
tan y = -^. Therefore 



= 12 
* ' 



20 



or 



d = 48. 



From triangle BOG, sec y = c/20, 
and, from Fig. 50, sec y = *-. 
Therefore 




13, 



_L = JJ1 

9.0 s > 



20 



or 



3 

FIG. 49. 

= (13) (20) = 
5 




From triangle ADC, 

DC = V~AC* + ZZf 2 



+ a 2 . 



22 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 
Replacing a, 6, and d by their values found above, we have 



FIG. 51. 



DC = V(48 + 21) 2 + 28 2 = 74.46. 
EXERCISES 



I- If, in Fig. 51, tan A = * and sec # = , find 
x, T/, and z. 




2. If, in Fig. 52, tan A 
find x, //, and z. 



= & and tan # - : f , 




3. If, in Fig. 53, sin A = f and tan B = 
find s, J, w;, x, y, and z. 




z 4. If, in Fig. 54, sin A = f and tan B 
find the lengths of all the line segments. 



FIG. 54. 



9] 



MISCELLANEOUS EXERCISES 



23 



5. Find the length of line segment y in 
Fig. 55. 




60V 



Fi. 55. 



6. Find length BD in Fig. 56.' 




60' 
Fm. 56. 




7. Find all unknown lengths of line 
segments in Fig. 57. 



Fio. 57. 
9. MISCELLANEOUS EXERCISES 

1. In each of the Figs. 58 and 59 read the six trigonometric functions 
of angle A. 




2. If sec A 



V29 



FIG. 58. 
" 




sm ^ cos 



Fio. 59. 
cot A. 



3. If sin A = f , show that 



(a) cos A cot A = T. (c) 1 + tan 2 A = sec 2 A. 

(b) sin 2 A + cos 2 A = 1. (d) 1 + cot 2 A = esc 2 A. 



24 TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE [CHAP. I 

4. Find the values of the trigonometric functions of an acute angle 
having (a) its sine equal to ; (b) its tangent equal to A; (c) its cosine 
equal to -J-l. 

5. If sin B , find the value of 



(a) 2 sin B cos #. 



(b) cos 2 J? - sin 2 B. 



6. If sin A = ^-, find sin 2A by means of the formula (to be derived 



later) 



sin 2A = 2 sin A cos A. 

7. If sin A = ^ and cos /? = f, find the value of sin (A + B) by 
means of the formula (to be derived later) 

sin (A + B) sin A cos B + cos A sin /?. 

8. The base of an isosceles triangle is 30 units, and each of its base 
angles has ^ as the value of its cosine. Find the lengths of the altitudes 
and of the sides of the triangle. 

9. For a certain triangle ABC, sin A = J-f , tan B = -V 5 -, and the 
altitude to side AB is 60 units. Find the lengths of the sides and of the 
altitudes of the triangle. 




100 
FIG. 60. 



10. Find all unknown line segments in Fig. 00 
if sin A = I, tan B = f . 




11. Find all unknown sides in radical form 
and all unknown angles in Fig. 61. 



FIG. 61. 



MISCELLANEOUS EXERCISES 



25 



12. In Fig. 62 tan a = f , sin 7 = , and 
sin j8 = Sff Compute the lengths of the 
sides of triangle ABC, and write the trig- 
onometric functions of angle ABC. 




IOV3 



Fio. 62. 



13. If, in Fig. 63, esc a = , AB = 29 units, 
BC = 25 units, and OD = 35 units, find the 
lengths of all line segments in the figure, and 
write the values of the trigonometric functions 
of 0, of 7, and of 6. Also find the length of 
the perpendicular from to the line DC. 




14. At a point A in a horizontal plane through the base of a flagpole 
the angle of elevation of its top is 35. If the flagpole is 40 ft. high, find 
the distance from A to the pole. 



16. In Fig. 64 CE is the median to side 
AB of the triangle ABC, tan A = i, 
AC = 37 units, and BD = 5 units. Find 
the lengths of all line se -nents in the 
figure, and write the trigoi metric func- A < 
tions of angle DCE. 



37 




E 
FIG. 64. 



B 5 



16. If, in Fig. 65, sin f , cos <p = f , 
AB = 20 ft., and CA = 16ft., find the lengths 
of all line segments in the figure. Also find the 
values of the trigonometric functions of angle 
AED. 




26 TRIGONOMETRIC Fl 'NCT1ON8 OF AN ACUTE ANGLE [CHAP. I 




17. In Fig. 66 ABC is an arc of a circle 
with center at O. Prove that angle DAB 
is 15. Compute the lengths DB, DA, 
and AB in radical form, and then write 
the trigonometric functions of 15. 

18. Construct a figure like Fig. 66 but 
with 45 in place of 30. Use the figure 
to find the trigonometric functions of 



FIG. 66. 



19. If the map distance B( 1 is 2.5 cm. (see Fig. 67) and if angle 
ABC = 25, find the map distance AB. 




FIG. 67. 

20. At a point midway between two trees on a horizontal plane the 
angles of elevation of their tips are 30 and 60, respectively. Show that 
one-iree is three times as high as the other. 

( 21yAn observer in an airplane (see Fig. 68) 2000 ft. above the sea 
sighrotwo ships A and B and finds their angles of depression to be 44 




2000' 



FIG. 68. 



and 32, respectively. If the observer is in the same vertical plane 
with the ships, find the distance AB (cot 44 = 1.036; cot 32 = 1.600). 



9] 



MISCELLANEOUS EXERCISES 



27 



22. The mine A in Fig. 69 is attached to 
the fixed point B by means of the 800-ft. 
cable AB. When the cable is vertical, the 
mine is 15 ft. below the surface of the water. 
How far from the surface is it when the tidal 
current has swung it to the position A' (cos 
38 = 0.788)? 




FKJ. 09. 



23. The ship represented in Fig. 
70 steams at a uniform speed due 
east. At 7 A.M. its captain ob- 
serves a lighthouse 10 miles away 
bearing due north, and at 7:30 
A.M. he finds that it bears 40 
west of north. Find the speed. 



24. Prove that the area K of a right 
triangle (see Fig. 71) may be exposed 




Fus. 70. 



X b = \ac cos ,4 = %bc sin .4 , 
tan A =* J a' 2 tan It, 

* * 1 : 2 sin /^ cos /?. 



i - __ 

/v = . 

7\ Jr' 2 sin A cos A = 




90VY 



b 
FIG. 71. 



CHAPTER II 



FUNDAMENTAL RELATIONS 
AMONG THE TRIGONOMETRIC FUNCTIONS 

10. Introduction. Since one value of a trigonometric function 
of an acute angle determines the angle and since there are six of 
these trigonometric functions, we naturally expect to find many 
relations connecting them. Among the forms of expressing a 
quantity there is usually one best adapted to our purposes. To 
obtain this one it is often convenient to use a number of elemen- 
tary identities. The main object of this chapter is to familiarize 
the student with these important elementary relations and give 
him the ability to use them with facility. 



write again the reciprocal relations 



11. Simple relations. For convenience of reference, we shall 

1 

A' 

(1) 



esc A = 



sin A 9 
1 



sec A = j 

cos A 

cot A = 7 r 

tan A 

Referring to triangle ABC in Fig. 1, we see that 



B 



-A 




\A 



tan A = 



a a/c __ sin A 
b b/c cos A' 

. A b b/c cos A 

cot A - -t-r = ' 7- 

a a/c sin A 



b 

FIG. 1. 



Therefore 

+o A 

tan A = 



- i 
cos A 



cot A = 



cos A 
sin A' 



(2) 



Another set of equations has reference to complementary angles. 
Referring to Fig. 1, we read from triangle ABC 

28 



11] SIMPLE RELATIONS 29 

sin A = - and cos (90 - A) = - 
c c 

Since sin A and cos (90 A) are both equal to a/c, we have 
sin A -= cos (90 - 4). 

By using the same kind of argument in connection with each of 
the trigonometric functions, the student may prove the following 
equations : 

cos (90 - 4) = sin A, sin (90 - A) = cos 4,) 

cot (90 - A) = tan A, tan (90 - A) = cot A>> (3) 

esc (90 - A) = sec A, sec (90 - A) = esc A,) 

or, stated in other words, any trigonometric function of an acute 
angle is equal to the co-function of its complement. This state- 
ment shows the significance of the prefix co- in the names of the 
trigonometric^ functions ; it has reference to the word complement. 
The relations (1), (2), and (3) are easily derived and recalled 
from a figure. First we construct Fig. 2 and from it read 

a A -A B 

- = sin A, or a = sin A, 

Y = cos A, or b = cos A. 

A' 

By replacing a by sin A and if; by cos A 

in Fig. 2, we obtain Fig. 3. Now apply 

the definitions of the trigonometric functions to read, from Fig. 3, 

A sin A , A cos A /ylN 

tan A = r> cot A = -p (4) 

cos A sin A 

A 1 A 1 

sec A = -p esc A = . v- 

cos A sin A 

Using (4) we obtain 

, , cos A . sin A 1 ,. 

COt A = -: j = 1 -5 j = 7 7' (6) 

sm A cos A tan A 

Next read the functions of (90 A) from Fig. 3 to get 
sin (90 - A) = cos A, cos (90 - A) = sin A, and the other 
relations of (3). Since one may obtain the relations (1), (2), and 
(3) directly from Fig. 3, it is only necessary to draw the figure to 
recall them. 





30 FUNDAMENTAL RELATIONS [CHAP. II 

12. Identities and conditional equations. An identity is an 
equation that is true for all values of the variables for which its 
members are defined. Thus the equations 

1 



esc x 



sin x 



are true for all values of x for which they are defined and are 
therefore identities. The equation x 2 = 1 is not an identity, 
since it is true only when x = 1 or 1. Similarly sin x = cos x 
is a conditional equation, since 45 is the only acute angle for 
which it is true. Equations (1), (2), and (3) of this article are, 
identities. Familiarity with these identities will be obtained 
by using them to simplify expressions, to verify identities, to 
find solutions of equations of condition, and to solve various 
kinds of problems. 

Example 1. Simplify 

sin A cos (90 - A) esc A cot A - sin (90 - .4). (a) 
Solution. From equations (3), we have 

cos (90 - A) = sin A, sin (90 - A) = cos A, (b) 
and from equations (1) and (2) 

.1 , A cos A f N 

esc A = . -p cot A = ~. ;r- (c) 

sm A sin A ^ ' 

Replacing cos (90 A), sin (90 A), cot A, and esc A in (a) 
by their values from (b) and (c), we obtain 

4 ' 4 * COS ^ A / IX 

sin A - sin A - r -- T cos A . (d) 

sin A sin A 

Since sin A is a number it may be canceled with sin A. Hence 
(d) simplifies to 

cos A cos A = 0. 

Example 2. Find an acute angle x which satisfies the equation 

sin (3x - 30) = cos (2z + 10). (a) 

* The symbol as is frequently used to mean "is identically equal to." 
However, for convenience, we shall use the ordinary symbol of equality 
throughout the book. 



12] IDENTITIES AND CONDITIONAL EQUATIONS 31 

Solution. Using the first equation of () to replace cos (2x + 
10) of (a) by sin (90 - 2x - 10), we obtain 

sin (3x - 30) = sin (90 - 2x - 10). 
This equation is satisfied if 

3 X - 30 = 90 - 2x - 10. 
Solving this equation for z, we get x = 22. 

EXERCISES 

1. Express as trigonometric functions of angles less than 45 

(a) sin 75. (c) tan 8930'. (e) cot 4550'. 

(b) cos 87. (d) see 4920'. (/) esc 7020'16' / . 

2. Find for each of the following equations an acute angle that satisfies 
it: 

sin (2x - 20) = cos (3x + 10). 
cos (50 - 10) = sin (30 + 20). 
tan (65 - 30) = cot (5 + 70). 
esc (20 + 70) = sec (40 - 36). 

3. Simplify 

(a) sin cot 0. 

(b) cos tan 0. 

(c) sec cot 0. 

(d) cos (90 - 0) sec cot 0. 

(e) esc cot (90 - 0). 

(/) sin cos (90 - 0) esc tan (90 - 0). 

(g) (tan 0) 2 (cos 0) 2 (esc 0) 2 . 

(h) (cot 0) 2 [cos (90 - 0)] 2 (sec 0) 2 . 

(i) sin cos (90 - 0) tan (90 - 0) (sec 0) 2 . 

4. Draw Fig. 3, and apply the definitions of the trigonometric func- 
tions to read from it all six functions of A and of 90 A. Compare 
the result with equations (1), (2), and (3). 

5. Verify each of the following identities by transforming the left-hand 
member, the right-hand member, or both members until they have the 
same form: 

(a) 1 + sin a cot a = sin a esc a + cos a. 

(b) tan a + sec a = sin a esc (90 a) + tan a esc a. 

(c) (sin a) 2 esc a cot a cos a = (cos a) 2 sec a tan a sin a. 

(<*) &S = (sin 0) 4 (sec 0) 2 (esc 0) 2 . 



32 



FUNDAMENTAL RELATIONS 



[CHAP. II 



(e) 



cot 6 



(/) cos ^ csc ^ tan <p = 1. 

(0) (sin A) 2 (csc A) 2 + (cos AY (sec A) 2 = 2. 
/IA cos A tan A , . . xo . 

W tan(90-A) = (sm ^ sec ^ 

(1) tan (cos 0) 2 - tan (90 - 0) (sin 0) 2 = 0. 
*tj) sin tan sec = sec cot (90 - 0) sin 6. 

(k) sec cot B cot (90 - 0) - sin csc (90 - 6) = sec - tan 0. 

tan 

(m) tan (30) tan (90 - 30) + sin (20) csc (20) + cos sec = 3. 

6. For each of the following equations find an acute angle that satisfies 
it: 

tan (60 - 50) tan (57 + 0) = 1. 

sin (90 + 1012 ; ) sec (20 + 840') = 1. 

csc (40 + 4329') cos (50 + 513 ; ) = 1. 

tan (80 - 35) sin (20 - 22) = cos (20 - 22). 

13. Relations derived from the Pythagorean theorem. From 
the right triangle ABC of Fig. 4 we have, by the well-known 
Pythagorean theorem, 



Dividing both members of this equation first by c 2 , then by b 2 , 

and finally by a 2 , we obtain 



(8) 




b 

Fio. 4. 



Expressing the quantities inside the parentheses in terms of 
trigonometric functions of the angle A, we have 

sin 2 A + cos 2 A = 1, ) 

tan 2 A + 1 = sec 2 A, > (9) 

1 + cot 2 A = csc 2 A, ) 

where sin 2 A means (sin A) 2 , cos 2 A means (cos A) 2 , etc. 



13] PYTHAGOREAN THEOREM 33 

Equations (1), (2), (3), and (9) should be memorized. 

Another method of deriving these 
formulas consists of applying the 
Pythagorean theorem to Fig. 5 to 



obtain ^ cosA 



sin .A 



sin 2 A + cos 2 A = 1 FIG. 5. 

and then dividing this equation first by cos 2 A and then by 
sin 2 A to obtain 

sin 2 A cos 2 A _ _ 1 _ 
cos 2 A cos 2 A cos 2 A' 
or 

tan 2 A + 1 = sec 2 A, 
and 

sin 2 A cos 2 A __ 1 
sin 2 A sin 2 A ~~ sin 2 A 9 
or 

1 + cot 2 A = esc 2 A. 

,. EXERCISES 
v 

1. By using relations (9) simplify 



1 

(b) 1 - cos 2 /?. (e) I - csc 2 ft. (h) 



_ t 
(c) sec 2 /? - 1. (/) csc 2 /3 - cot 2 j8. 

2. Use equations (1), (2), (3), and (9) to simplify 

sin 2 <p + cos 2 <p 
(a) ~1^V> cos ? ' (d) tan * + cot v - 

(6)(s e c 2 ,-l)(cscV-l). 
. . (1 sin ^)(1 + sin >p) .. 
' 



(l-eoBv>)(l+coB) 

3. Transform each of the following expressions so that the equivalent 
expression will contain only sines and cosines of 6, then replace cos 6 by 
\/i sin 2 so that the final expression will contain no trigonometric 
functions except sin B: 

(a) 2 sin 6 cos 4 B tan 2 B. (d) (tan B - cot 6) sin B cos B. 



(b) -2 ' (*) sec * ~ sin2 sec2 



(c) cos 4 B - sin 4 0. (/) tan sec 2 - cot (90 - B). 



34 FUNDAMENTAL RELATIONS [CHAP. II 

4. Transform each of the expressions in the left-hand column into the 
one written to the right of it. 

(a) esc 2 + sec 2 6 sec 2 esc 2 



_ _ 
tan 2 A +"l "" cot 2 A + I 

(c) cos tan sin 

(d) sin 2 -f- esc 2 sin 4 

cot2 A 



(/) cos 2 A tan 2 A + sin 2 A cot 2 A 1 

, . 1 , tan 2 A . 

( )1 + secA 



14. Verification of identities. There are two methods of 
procedure for verifying identities. By means of the fundamental 
identities* and suitable algebraic operations, (a) the more com- 
plicated member of the identity may be transformed into the other 
member of the identity; (b) both members may be transformed into 
the same expression. II may be advisable, as a last resort, to trans- 
form both members into expressions that contain only one trigo- 
nometric function. The following examples will illustrate methods 
of procedure: 

Example 1. Verify the identity 

(tan + cot 0) 2 = sec 2 + cse 2 0. 
Verification. Expansion of the left-hand member gives 

tan 2 + 2 tan cot + cot 2 0. 
Since col tan = 1, we may write this in the form 

(tan 2 + 1) + (1 + cot 2 0). 

From the last* two equations of (9), this expression is 

see 2 + cse 2 0. 

* Although we have proved the identities (1), (2), (3), and (9) only for 
acute angles, they will he found to he true, as soon as we have defined the 
trigonometric functions of the general angle, for all angles for which the 
functions are defined. A similar statement applies to all the identities of 
this article. 



14] 



VERIFICATION OF IDENTITIES 



35 



2 esc 2 - esc 4 

2 1 

sm 2 



Example 2. Verify the identity 

1 cot 4 = 2 esc 2 - esc 4 0. 

Verification. In the following outline, the work on the left 
of the vertical line gives the steps for reducing the left-hand 
member to a function of sin 0; the work on the right of the 
vertical line applies to the right-hand member: 

1 - cot 4 = 

i cos4 _ _j^ i_ 

sin 4 ~ sin 2 !? sin 4 

sin 4 cos 4 0* _ 2 sin 2 01 

sin 4 ~~ sin 4 

sin 4 - (1 - sin 2 0) 2 = 
sin 4 ~~ 

-1+2 sin 2 
sin 4 

Thus the identity is verified^ since we have shown that both its 
members are equal to the same expression. 

Alternative verification. The steps outlined in the following 
plan give a more direct verification: 



1 _ cot 4 = 

(1 + cot 2 0)(1 - cot 2 0) = 
esc 2 0(1 - cot 2 0). 



2 esc 2 - esc 4 = 

esc 2 0(2 - esc 2 0) = 

esc 2 0(2 - cot 2 - 1) = 

esc 2 0(1 - cot 2 0). 



EXERCISES 

Simplify each of the following expressions: 

1. tan x sin x + cos x. 

2. cot A sec A esc A(l 2 sin 2 A). 

3. (tan B + cot B) sin B cos B. 

4. tan A sin A cos A + sin A cos A cot A. 
6. (cot 2 A - esc 2 A)(sec 2 A - tan 2 A). 

6. (cos 2 0-1) esc 2 0. 

Transform each of the following expressions into the expression written 
to the right of it: 

* Beginning at this point we could have written 
(sin* - cos 2 flXsin 1 e + cos 2 0) = sin 2 (1 sin 2 e) 2 sin 2 - 1. 



36 FUNDAMENTAL RELATIONS [CHAP. II 

7. cos csc 6 tan 0. 1. 

8. tan A sec A cot A cos A tan (90 A). cot A. 

9. esc A cot A cos A + 1. esc 2 A. 

10 ' sin^A + c^T sec2 A csc2 A - 

11. sec 2 A csc 2 A. sec 2 A + csc 2 A. 

12. (sec cos 0)(csc sin 0). sin cos 0. 

13. (sec A - tan A) (sec A + tan A). 1. 

14. (csc A cot A) (csc A + cot A). 1. 

16. sin (90 - B) cot B sin B - 1. - sin 2 B. 

16. 2 cos 2 A - 1. 1-2 sin 2 A. 

17. sec 2 A + tan 2 A. 2 sec 2 A - 1. 

Verify the following identities: 

18. sin sec cot = 1. 

19. (tan y + cot y) cot y = csc 2 #. 

_ _ . . sec A 

20. tan A = - j- 

csc A 

21. (cos A - l)(cos A + 1) = - sin 2 A. 

22. cot C sin C + cos C = 2 cos C. 

23. tan (90 - A) tan A - cos 2 (90 - A) = sin 2 (90 - A). 

24. sin cot + cos 2 sec = 2 cos 0. 

25. cos 2 a(l + tan 2 a) = 1. 

26. cot cos + sin = csc 0. 

27. sin 2 A sec 2 A = sec 2 A 1. 

28. (sin <p cos <p) z = 1 2 sin <p x>s ^. 

cos 8 , cos 

29 -r+^ + r^E^ = 2sec ^ 

30. sin 4 x cos 4 x = 2 sin 2 z 1. 

31. (1 - sec 2 A)(l - csc 2 A) = 1. 
00 1 + tan 2 a 



cos* 



>Ol+sin*^ cos* ---* 

34. csc 2 (? csc 2 <p cos 2 (p = 1. 

35. tan * + cot x - sec x csc x. 

36. (cot a. tan a) 2 sin 2 a cos 2 a =* 1 4 sin 2 a cos 2 a. 

37. sec 4 a - tan 4 a sec 2 a + tan 2 a. 



15] 



FORMULAS FROM RIGHT TRIANGLES 



37 



38. r 



sec A + esc A 

~ = sec A csc 



39. 

40. 
41. 

42. 
43. 



cot 6 



csc + 1 = 

cot ~~ csc 1 
tan A sin A + cos A = sec A. 
csc 4 A - cot 4 A = 2 cot 2 A + 1. 
tan x cot x 



sec x + csc x. 
sin 



sm x cos x 
tan sin __ 
tan sin ~~ 1 cos 
cot B cos B 1 sin 



- - \s\jv -LJ v/vyo JL-f 

** ~~ a o ' 



45. 
46. 

47. 
48. 
49. 



cos 2 B sin 7? 
tan <p csc ^? sec ^>(1 2 cos 2 ^>) = cot 
cos 6 A + sin 6 A = 1 3 sin 2 A cos 2 A. 



v 



1 cos 



1 r- 

1 + COS 



, 

= CSC X COt X. 



/sec v? tan 
'sec <p + tan 
sec y + tan T/ 



A 
= sec # tan 



60. 



c . sm y 

61. cot w + v , = csc i/. 

y 1 + cos y 

cos A , 1 sin A _ , 

62. t ,-::--7 + 4 = 2(sec 



i 



64. 



_____ 

(cos 2 i~ 13n*")" 2 " 
tan 6[_+ sec^6>_-- 1 
tan sec + i 



cos 



- tan 

_ 

= 



15. Formulas from right triangles. It appeared in 11 that we 
could read formulas (1), (2), and (3) directly from Fig. 3. Other 
identities may be obtained in the same manner. 

For example, we draw the right triangle 
shown in Fig. 6 with leg AC equal to 1. 
Then 

Y = tan A y 
T = sec A. 




38 



FUNDAMENTAL RELATIONS 



[CHAP. II 



Figure 7 is obtained by replacing a by tan A and c by sec A 
in Fig. 6. Using the definitions of the trigonometric functions 
on Fig. 7, we get 



AC 
CB 



cot A 
cot (90 - A) = ^ 



7 - T 

tan A 
tan A, 



COS ^ = 



AC 



-7-77 - 

AB sec 
esc (90 - A ) = 



sec A. 




By applying the Pythagorean theorem 
to Fig. 7, we got 



1 + tan 2 A = sec 2 A. 



(10) 



sin A 



tan 



sec A 



Evidently other identities could also be 
obtained. Thus, from Fig. 7, we read 

fi\t\Q A \ tan -ri 

cos (90 A} = -T-, etc. 

7 sec A 



Figure 8 was obtained by using the idea underlying the construc- 
tion of Fig. 7. From it we read 



B 




tan A 



1 
cot A' 



sin A = 



esc A 



cot A 

Fiu. 8. 



1 tan B = tan (90 - A) = cot A, 
. sec (90 - A) = rsc A, 

C 

1 + cot 2 A = esc 2 A, 



(ID 



and others. The fundamental identities can be recalled at any 

time by reproducing Figs. *3, 7, and 8 and reading the identities 

directly from these figures. 

By means of figures, it is a simple matter to express all of the 

trigonometric functions in terms of one. Figure 9 is about the 

same as Fig. 7; instead of replacing 
AB by sec A, we have observed that 




1 

FIG. 9. 



AB = 



+ m* = Vl + tan 2 A 



and have written \/l + tan 2 A on A B. 
The definitions of the trigonometric, 
functions may now be used to read from Fig. 9 



515] 



sin A = 



FORMULAS FROM RIGHT TRIANGLES 

tan A J 1 



COS A = - 



+ tan 2 

7 - A 

tan A 



+ tan 2 A " Vl + tan 2 

sec A = Vl + tan 2 A, G SC A = 

1 

cot A = 7 T- 

tan A 

EXERCISES 



1. Using Fig. 10, express all the trig- 
onometric functions of angle A m terms 
of sin A. ^ 



sin A 



VT-sin 2 ^ 
FIQ. 10. 



2. Using Fig. 11, express all the trigonometric 
functions of angle .1 in terms of cos A. 



COB A 

FIG. 11. 



3. Express all the trigonometric functions of angle A in terms of (a) 
cot A, (6) sec A, (c) esc A. 



4. In Fig. 12 AC = 1. Find the 
lengths CB, AB, AD, and DC and 
equate two values of AC to obtain 

sin 2 A + cos 2 A = 1. 




FIG. 12. 



6. In Fig. 13 AD = 1. Find 
the lengths of AB, BD, AC, and 
CD and equate two values of AC 
to obtain 

1 + tan 2 A = sec 2 A. 




Fio. 13. 




FUNDAMENTAL RELATIONS 



[CHAP. II 



6. In Fig. 14 BC = 1. Find AB, BD, AC, and 
CD and equate two values of BD to obtain 



FIG. 14. 




1 + cot 2 



esc 2 A. 



FIG. 15. 



7. The radius of the circle in Fig. 15 is 1. 
Find the lengths of the line segments PQ, 
OQ, TD, OT, OC, BC, write them on the 
figure, and read from the figure the follow- 
ing identities: 

sin 2 A + cos 2 A = 1, 
1 + tan 2 A = sec 2 A, 
1 + cot 2 A = esc 2 A. 



16. Length of line segments. The same ideas employed in 7 
may be used in connection with more complicated figures. The 
ability to express all parts of a rectilinear figure simply in terms 
of given parts is one of the most important values obtained from 
a study of trigonometry. It enables one to derive and recall the 
important formulas of trigonometry and to derive simple for- 
mulas for heights and distances. 

Consider the right triangle ABC shown in Fig. 16. The given 
parts A and c are encircled. First let us try to express x, h, y, 




FIG. 16. 



16] 



LENGTH OF LINE SEGMENTS 



and a in terms of the given parts. From triangle ABD, we 
write 



- = cos A ; 



/. x = c cos A. 
.'. h = c sin A. 

Similarly, from triangle BDC, we have 

v 

j- = tan A ; :. y h tan A. 



. , 
- = sm A ; 



(12) 
(13) 

(14) 



Replacing h in this formula by its value c sin A from (13), we 
have 

y = c sin A tan A. (15) 

Also from triangle BDC, we get 



** A 

T = sec A ; 



/. a = h sec - 



(16) 



Replacing h in this formula by its value c sin A from (13), we have 
a = c sin A sec A. (17) 

Figure 17 is obtained from Fig. 16 by replacing x, y, h, and a by 
their values from (12), (14), (13), and (17), respectively. 

B 




FIG. 17. 

It is to be observed that when there are given only enough 
parts of a rectilinear figure to determine it and when all parts 
of the figure have been expressed in terms of the given ones, 
then any relation obtained by reading an equation from the 
figure, either by applying a proposition from geometry or by 
using the definitions of the trigonometric functions, is an identity. 
Thus an identity may be formed from Fig. 17 by using the 
Pythagorean theorem. In accordance with it, 

(18) 



42 



FUNDAMENTAL RELATIONS 



[CHAP. II 



Replacing the lengths of the line segments in (18) by their values 
from Fig. 17, we get the identity 

c 2 + c 2 sin 2 A sec 2 A = (c cos A + c sin A tan A) 2 . 

That this is an identity may be verified in the usual way. 

The student will find the following statement helpful while ho 
is becoming familiar with the method. 

To find the lengths of line segments of a rectilinear figure in 
terms of specified parts and to obtain identities: 

(a) Draw a figure, encircle each symbol representing a specified 
part, and put a letter on each of the other parts. 

(b) Find all angles of the figure in terms of encircled angles. 

(c) Use the definitions of the trigonometric functions to express 
all parts in terms of specified parts. 

(d) Form identities by using the definitions of the trigonometric 
functions, by equating two expressions for the same length or area, 
and by using theorems from geometry. 

EXERCISES 

1. In Fig. 18 show that AB = sec 
A, BD - tan A, BC = tan A sec A, 
DC = tan 2 A. Write each of these 
values on the appropriate line of the 
, figure and then apply the Pythagorean 
theorem to triangle ABC to obtain an 
identity. 

2. In Fig. 19 find DE and CE in 
terms of a, A, and B. 

Hint. Find in order the lengths 
E DF, DE, FE, CF, CE. 




'A 



1 D 




3. In Fig. 20 find the length of OE. 
Hint. Find in succession the lengths OB, 
OC, OD, and OE. 



16] 



LENGTH OF LINE SEGMENTS 



43 



4. In Fig. 20 replace 8 by (90 - 6), and then find the length of OE 
in the resulting figure. 



6. Compute the lengths of AB and AD in 
. 21. 




6. Compute lengths FE and BC 
in Fig. 22 (angle ABE * 90). 

Hint. To find the length of BC, 
find in succession the lengths x, y, 
BC. 




47 




B 
FIG. 22. 



7. In Fig. 23 find the lengths DC, BC, 
and A#, and then read from the figure a 
formula for tan ?6 in terms of sin 6 and 
cos 0. 



B 



FIG. 23. 



D 




8. In Fig. 24 AB is parallel to DE. 
Find A# and DE in terms of a and 0. 

//i'nt. Find in succession the lengths 
CB, AB, DB, DE. 




FIG. 24. 



44 



FUNDAMENTAL RELATIONS 



[CHAP. II 




9. In Fig. 25 find in succession the lengths 
ED, FE, FD, AD, CD, AC in terms of and 
<p, and write each of them on the appropriate 
line segment of the figure. 



10. In Fig. 25 erase 1 from AE, take 
AC = 1, and find in succession the lengths 
CD, AD, DE, FE, FD. 



FIG. 25. 



11. Draw an isosceles triangle with vertical angle equal to 20; drop a 
perpendicular from the vertical angle to the side opposite and a perpen- 
dicular from a second angle to the side opposite. Find the values of all 
line segments in the figure thus drawn. Write two expressions for the 
area of the triangle and equate them to obtain an identity. 

17. MISCELLANEOUS EXERCISES 
1. Express as trigonometric functions of angles less than 45: 



(a) sin 65. 
2. Simplify: 



(6) tan 49. 



(c) sec 82. 



(a) cot tan (90 - 0) sin 2 0. 

(6) sin tan cos 9 + cos 2 0. 

(c) (sin + cos 0) 2 + (sin - cos 0) 2 . 

(d) sin esc + tan 2 0. 



(f) cot (90 - 0) sin cos 0. 

(g) cot (90 - A) - tan A + sin 90 + tan 45. 

3. Transform each of the expressions in the left-hand column into the 
one written to the right of it. 



(a) sin cot 6. 

(b) sin sec 0. 

cos 2 A 



(d) 



esc 2 0-1 
sec* 0-1* 



COS 0. 

tan 0. 

1 + sin A. 

cot<0. 



17] MISCELLANEOUS EXERCISES 45 



v * y sec A cos A 
1 + sin A 1 sin A 


cut .0. UBU A. 

4 tan A sec A. 

csc 2 A + cot 2 A. 
sin 0. 

tan 2 A. 
tan A cot A. 
2(1 + tan*0). 


^' 1 sin A 1 + sin A 
(0) csc 4 A cot 4 A. 


(A) cos 0Vsec 2 1. 
1 + sin 2 A sec 2 A 


W 1 + cos 2 A csc 2 A 
1-2 cos 2 A 


'" sin A cos A " 
1 + cos A 1 cos A 


^^ sen A tan A sen A 4- tan A 



4. Express each of the following in terms of sin A : 

(a) cos A cot A. , (c) tan A /sec A. 

(6) sin A(cot 2 A + 1). ' 4 (d) cos 4 A - sin 4 A. 

5. Express each of the following in terms ^f cos A: 

(a) sin A cot A. (6) cot 2 A/(l + cot 2 A). 

6. Express each of the following in terms of tan 0: 
(a) (sec 2 - 1) cot 0. . (6) sec 4 - sec 2 0. 

7. Change each of the following to equivalent forms involving only 
sin and cos 0: 

(a) tan + cot 0. (6) esc - cot 0. (c) sec + tan 0. 

X 2 7/ 2 

8. (a) If x = a cos and y = b sin 0, show that - 2 + ^ = 1. 

x 2 i/ 2 
(6) If x = a sec and y = b tan 0, show that -j ^ = 1. 

(c) If x a cos 3 and y = a sin 3 0, show that xf + 2/ = o^. 

9. In each of the expressions in the left-hand column replace x by its 
value written opposite, and solve the result for y: 

x 2 + y 2 a 2 . x = a cos 0. 

6V + ay = a 2 6 2 . a; = a cos 0. 

(c) 6 2 x 2 - ay = a 2 6 2 . x = a sec 0. 

(d) zi + ?/i = ai. a; = a cos 4 0. 

(e) zi + i/i = ai. x = a cos 6 0. 
(/) zy = 6 2 x 2 + ay. a? = a sec 0. 

(0) x z y* = ay 6 2 x 2 . x = a sin 0. 
(h) y*(2a - x) = x\ x = 2a sin 2 0. 

(1) 2/ 2 (* 2 + 4a 2 ) = 16a 4 . x = 2a tan 0. 



46 



FUNDAMENTAL RELATIONS 



[CHAP. II 



Verify the identities numbered 10 to 37. 

10. sec x cos x = sin x tan x. 

11. tan 2 x csc 2 x cot 2 a: sin 2 x = 1. 

12. tan 2 x cos 2 a: + sin 2 x cot 2 a; = 1. 

13. (1 + tan 0)(1 + cot 0) sin cos = 1 + 2 sin cos 0. 

14. (tan + cot 0) 2 = sec 2 csc 2 0. 
16. sec 2 x + csc 2 x = sec 2 a; csc 2 x. 

16. sec 4 x sec 2 x = tan 4 a: + tan 2 x. 

17. sin cos 0(sec + csc 0) = sin + cos 0. 

18. sin 2 x sec 2 a: = sec 2 x 1. 
1 + tan 2 A _ sin 2 A 

1 + cot 2 A = cos 2 !!" 
sin A , cos , ' 



19. 
20. 



csc A 
</21. cot A 



sec 



1. 



sin A 



1 + cos A* sin A 
22. sec 4 0-1 = 2 tan 2 + tan 4 0. 
csc 



23. 



cot + tan 
24. (tan + sec 0) 2 



= cos 0. 



/I +sin0y 
\ cos / ' 



cos 

1. sin x(l + tan a:) + cos x(l + cot a-) 
sin a; , 1 + cos x 



sec r + csc 



1 + cos x sin a; 

cos sin _ 

F^tan "*" 1 - cot " 

- cos (1 - cos 0) 2 



2 csc x. 

sin + cos 0. 



3/1 +COS0 

sec a 



sin 2 
tan x sin a? 



1 + cos x 

cot x + csc x = y 

sin 3 + cos 3 



sin x(l cos 2 x) 
sin x 



; a- 1 sin cos 0. 

sin + cos 

2. sec 6 - tan 6 = 1+3 sec 2 tan 2 0. 

33. cos 6 A - sin 6 A = (2 cos 2 A - 1)(1 - sin 2 A cos 2 A). 

34. (cos 2 x - l)(cot 2 x + 1) + 1 = 0. 

>/35. 2(sin 6 + cos 6 0) - 3(cos 4 + sin 4 0) = -1. 
tan 2 + cot 2 = sec 2 csc 2 0-2. 
I sec 2 + cos 2 = tan 2 sin 2 + 2. 



17] 



MISCELLANEOUS EXERCISES 



47 



38. In Fig. 26 compute the length of x. 




20 
FIG. 26. 



39. Compute the lengths of A B and A D in 



40. Compute the length of each line 
segment in Fig. 28. 




41. In Fig. 29 compute y by first 
finding x. 



42. In Fig. 30 find the lengths of 
A C and A Bin terms of a, 0, <, and 
a. 




Fia. 30. 



CHAPTER III 




GENERAL DEFINITIONS 
OF TRIGONOMETRIC FUNCTIONS 

18. Definition of angle. Only trigonometric functions of 
angles no greater than 90 have been considered in the first two 
chapters. This chapter will be concerned with functions of 
angles that may have any magnitude. 

A half line or ray is the part of a straight line lying on one side 
of a point of the line. It is designated by naming its end point 

and another point on it. Thus OA in 
Fig. 1 is the ray beginning at O and 
extending through A. If a half line or 
ray beginning at point rotates about 
O in a plane from an initial position OA 
to a terminal position OB, it is said to 
generate the angle AOB (see Fig. 1). When the legs of a 
compass are drawn apart an angle is generated; the hands of 
a clock rotate arid generate angles. 

When the generating ray is turned through one-fourth of the 
complete turn about a point, the angle generated is called a right 
g Q angle; a degree is - 9 1 - of a right 

angle, a minute is -^ of a de- 
gree, and a second is -^ of a 
minute. Although either di- 
rection of rotation may be con- 
sidered positive, it is customary 
in trigonometry to call angles 
generated by counterclockwise 
rotation positive angles and 



Counter clockwise 

or positive rotation 

(a) 

FIG. 2. 



Clockwise or 

negative rotation 

(6) 



those generated by clockwise 
rotation negative angles. In Fig. 2 (a) the curved arrow indicates 
counterclock-wise or positive rotation through five right angles; 
in Fig. 2(5) a negative right angle is indicated. 

48 



19] 



RECTANGULAR COORDINATES 



49 



EXERCISES 

1. Construct the following angles: 



(a) 6 right angles. 

(b) 6 right angles. 

(c) 5 right angles. 



(d) 3 right angles. 

(e) 3| right angles. 
(/) -2i right angles. 




2. Through how many right angles does the 
minute hand of a clock turn from 12:15 P.M. to 2 
P.M. of the same day [see Fig. 3 (a)]? 



FIG. 3a. 

3. What are the magnitude and sense of the angles generated by the 
hour hand of a clock between 3 A.M. and the next 8 A.M.? 

4. Through what part of a right angle does the minute hand of a clock 
move in 1 min. of time? 



5. A Ferris wheel is turning through 3 revolutions 
in each minute. Through how many right angles will 
it turn in 2 min. [see Fig. 3(6)]? 



FIG. 36. 

6. An imaginary line connecting the center of the earth's orbit to the 
center of the earth makes one complete revolution each yeai . Assuming 
that this line turns in a plane at a constant rate, find the number of right 
angles described by this line in (a) 3 months- <b) 7 months; (c) 25 months; 
(d) 2000 years; (e) 1 day; (/) 1 hr. 

19. Rectangular coordinates. This article is designed to recall 
the essential conceptions of rectangular coordinates; they are used 
in the definitions of the trigonometric functions of any angle. 

In Fig. 4, X'X represents a straight line, and is any point on 
it. If we choose a unit of measure, any point to the right of 
will be designated by a positive number telling its distance from 
O in terms of the chosen unit, and any point to the left of will 
be designated by a negative number whose magnitude gives the 
distance of the point from 0. Thus a point 5 units to the right 




50 



GENERAL DEFINITIONS 



[CHAP. Ill 



of is designated by 5, whereas a point 3^ units to the left of O 
is designated by 3.5. 



O 



-7 -6 -5 -4 



-3 -2 -1 1 

FIG. 4. 





^<S r 


^ V 


Ill 


O 


* Q '" 
IV 



Y 1 

Fin. 



By means of a system called rectangular coordinates, the 
position of any point in the piano is defined by two numbers. 
In this system two mutually perpendicular lines, referred to 
as axes, are required. In Fig. 5, X'X and Y'\ represent two 
perpendicular lines intersecting at 0. The four parts into which 
the plane is divided by these lines are culled the first, second, 
v third, and fourth quadrants, respec- 

tively, as indicated in the figure. Let 
P be any point in the plane of X'X and 
Y'Y. Drop a perpendicular from P to 
the .r-axis, meeting it in Q, and another 
from P to the */-axis, meeting it in R. 
Lot x, considered as positive when P is 
to the right of Y'Y and as negative 
when P is to the left of Y'Y, be the 
measure of OQ in terms of a given unit of measure; let y, con- 
sidered as positive when P is above X'X and negative when P is 
y below X'X, bo the measure of OR 

in terms of the given unit. Then 
any point in the plane will be rep- 
resented by a pair of numbers, x 
and y. 

The first number x is called the 
abscissa of the point P, and the 
second number y is called its ordi- 
nate. The two numbers x and y 
are called the coordinates of P, and 
the point is designated (x, y). 
Thus in Fig. 6 the abscissa of PI is 
2, its ordinate is 3, its coordinates are 2 and 3, and it is 
designated (2, 3). Similarly, P 2 is designated (3, 3), Pa 
is designated ( 2, 3), and P 4 is designated (1, 2). 

EXERCISES 

1. Plot the points (2, 4), (-2, 4), (2, -4), (-2, -4), (4, 2), (4, -2), 
(-4, 2), (-4, -2). Why do all these points lie in a circle? 



X'- 



O 



Y' 

FIG. 6. 



20] 



DEFINITIONS OF TRIGONOMETRIC FUNCTIONS 



51 



2. Plot the points (0, 1), (0, 5), (1, 0), (5, 0), (0, -1), (0, -5), (-1, 0), 
(-5,0), (0,0). 

3. Read the trigonometric functions of the angle subtended at by 
the line connecting (a) (12, 0) to (12, 5); (6) (x, 0) to (x, y), assuming 
x and y to be positive numbers. 

4. Where are all the points for which (a) x = 3? (b) y = 3? 
(c) x = -4? (d) y = 5? (e) x = 0? (/) y = 0? fo) r = 3? 

6. What is the abscissa of all points on the ?/-axis? What is the 
ordinate of all points on the z-axis? 

6. Determine the quadrant in which (a) the abscissa and ordinate are 
both positive; (6) the abscissa is negative and the ordinate is positive; 
(c) the abscissa is positive and the ordinate is negative; (d) the abscissa 
and ordinate are both negative. 

7. Assuming that r is always positive, in which quadrants are each of 
the following ratios positive? in which negative? 

(a) y/r. (b) x/r. (r) x/y. (d} y/x. (e) r/x. (/) r/y. 

20. Definitions of the trigonometric functions of any angle. 

Appropriate definitions of the trigonometric functions of any 
angle are desired. Consider the obtuse p Y 

angle XOP in Fig. 7. The point P on 
the terminal side of the angle has 
coordinates x = 3 and y = 4 as 
shown. Evidently OP 5 is the hy- 
potenuse. Previously the side along 
the initial line was called the adjacent 
leg. Hence OR = x = 3, the initial 
line produced, should be called the 
adjacent leg. Also, RP = y = 4 does not lie along a side of the 
angle and should be called the opposite leg. 

Therefore, using the definitions previously given in 3 and 4, 
we would naturally write 





\ 






\5 




4 




A 




-, \ 


\ 


R -3 


* 


Fir,. 7 





. A __ opposite leg _ 4 

sin /i i > 

hypotenuse 5 

4 adjacent leg 3 

cos A = T-J = -> 

hypotenuse 5 

tan A = PP site le g = _i_, 
adjacent leg 3' 



, hypotenuse 5 

CSC A = TT j = 7? 

opposite leg 4 

. hypotenuse 5 

sec A = j^ -, = ^> 

adjacent leg 3 

, 4 adjacent leg .3 
cot A = --- .. . & = --=- 
opposite leg 4 



In Fig. 8, the coordinates of P are x = 3, and y = 4. 
Calling 5 the hypotenuse, 3 the adjacent leg, and 4 the 



52 



GENERAL DEFINITIONS 



[CHAP. Ill 



opposite leg, we would naturally write in accordance with the 
definitions of 3 and 4 




sin A = 



-4 
5 ' 
-3 



csc A 


u 






5 




sec A 


- g> 




cot A 


-3 
-4 


3 
4 



FIG. 8. 



tan A = 



In Fig. 9 the coordinates of P are 
x = 3, y = 4. Taking 5 as hypotenuse, 
a; = 3 as adjacent leg, and y = 4 as opposite leg, we write, in 
accordance with the definitions of 3 
and 4, 




sin A 


4 
= ~5-' 


csc A 


5 
~ -I' 




_ 3 




_ 5 


cos A 




sec A 






5 




3 


tan A 


_ -4 


cot A 


_ 3 




"~ 3 ' 




-4* 



FIG. 9. 

The foregoing discussion suggests definitions of the trigono- 
metric functions of any angle. Draw the axes for a set of rec- 
tangular coordinates and consider the angle A generated by a 
ray in turning about the origin from the positive x-axis as 
initial position to any terminal position. Let P be a point on 
the terminal ray, let r, considered as positive, be the distance 
along this ray from to P, lot x be the abscissa of P and y its 
ordinate, as shown in Figs. 10(a), (6), (c), (d). We then define 
the trigonometric functions of angle A as follows: 



. 
sin A = 



ordinate 
-JT - 
distance 



. abscissa x 
cos A = -J7-T - = - 
distance r 



tan A 



ordinate _ y 
abscissa x 



A distance r 

CSC A = -j: T- = - 

ordinate y 
distance _ r 
abscissa ~~ x 
abscissa x 



. 
sec -4 = 



cot A = 



ordinate 



The student will perceive lhat the definitions (1) are natural 
extensions of the definitions given in 3 and 4 if he will associate 
side adjacent with abscissa x, side opposite with ordinate y, and 



20] DEFINITIONS OF TRIGONOMETRIC FUNCTIONS 53 



hypotenuse with distance r. Note that the definitions (1) include 
as a special case the definitions given in 3 and 4. 



*(-) 
(6) 










FIG. 10. 
EXERCISES 



1. Read the values of the trigonometric 

functions of an angle A if its cosine is $ and * 

(a) if it is a second-quadrant angle (see Fig. 11); ^ 

(6) if it is a third-quadrant angle. 




O 



->X 



FIG. 11. 



2. Write the appropriate signs, + or , in the blank spaces of the 
following form : 





sin 


cos 


tan 


cot 


sec 


CSC 


1st quad 


-fr 


-*- 


-V 


*- 


-V 


-v 


2d quad 


H- 














-V- 


3d quad 








f 


-f- 








4th quad 





-V 








-*- 






54 



GENERAL DEFINITIONS 



[CHAP. Ill 



V3 

3. The sine of a certain angle is $, and its cosine is jr Find the 

values of the other trigonometric functions of this angle. 

4. Fill in the blank spaces of the following diagram: 



Angle 


sin 


cos 


tan 


cot 


sec 


CSC 


A 


* 


iV3 










A 






1 




-V2 




A 








-V/3 




-2 


A 


A 


-H 










5. The absolute value (numerical value without reference to sign) of 



the tangent of an angle is 3^. Write the values of the six trigonometric 
functions of this angle (a) when it is less than 90; (b) when it is greater 
than 90 but less than 180; (c) when it is greater than 180 but less than 
270; (d) when it is greater than 270 but less than 360. 

6. Each of the following points is on the terminal side of an angle 0, 
in standard position; find the trigonometric functions of 0. 

(a) (4, 3). (d) (15, -8). (g) (2, -3). 

(b) (-4,3). (e) (24, -7). (A) (1, V3). 

(c) (-5, -12). (/) (1,3). (i) (-2,0). 

7. In what quadrants may terminate under the following con- 
ditions: 



19. 

(a) sin pos.? (c) tan pos.? 
(6) cosflneg.? (d) cot neg.? 



(e) sec neg.? 
(/) esc 0pos.? 



8. In what quadrant must terminate under the following con- 
ditions: 



(a) sin pos. and cos neg. ? 
(6) tan neg. and sec pos. ? 
(c) cot neg. and cos pos.? 



(d) cos rieg. and sin neg.? 

(e) cos neg. and esc pos.? 
(/) cot neg. and esc neg.? 



9. Locate the terminal side ofj? and find its other functions, having 
given: 



(a) cos = 
(c) sin = 



f , sin pos. 
l THt sin neg. 
TT, cot neg. 



(d) sec 

(e) esc : 
(/) cot 



, tan neg. 
~TT, tan pos. 
-T3, esc neg. 



22] VALUES OF TRIGONOMETRIC FUNCTIONS 55 

(g) sin = |, cos neg. 0") cot 6 = -, sin 6 neg. 

(A) sec = -2, sin neg. (k) cos = A, cot neg. 
(i) tan = -A, sec pos. (J) esc = -2, tan neg. 

10. Find the value of 2 tan 0/(l - tan 2 0) when cos = -f and 
is in the third quadrant. 

11. Find the value of (esc cot 0)(sin 2 + cos 2 0) when sec = 
f and tan is negative. 

12. If sin = f , find the values of (cos - esc 0)/cot for the 
various quadrants in which may terminate. 

21. Observations. We have seen in 3 and 4 that each of 
the six trigonometric functions of an acute angle has only one 
value. Similarly, each of the trigonometric functions of an 
angle, unrestricted in magnitude, has only one value. However, 
the converse is not true. Since the trigonometric functions are 
defined in terms of values dependent on an initial ray and a 
terminal ray, each of them has the same value for a given angle 
as for any other angle having the same initial position and the 
same terminal position as the given angle. In other words, 
the value of any trigonometric function of a given angle is equal to the 
value of the same trigonometric function of any angle differing from 
the given one by a multiple of 360. Hence, in finding the value 
of a trigonometric function of any angle, one may add to the 
angle or subtract from it any integral multiple of 360. 

Observing that x is negative and that y and r are positive in 
the second quadrant, we see that the sin (y/r) and esc 
(r/y) are positive and the other four trigonometric functions are 
negative for second quadrant angles. Similarly, x and y are 
both negative in the third quadrant, so that the tangent (y/x) 
and the cotangent (x/y) are both positive, and the other functions 
are negative for third quadrant angles. Finally, in the fourth 
quadrant, x and r are positive, so that the cosine (x/r) and the 
secant (r/x) are positive and the other functions are negative 
for fourth quadrant angles. 

22. Values of trigonometric functions for special angles. In 

5 (Chap. I) we were able to read from appropriate figures the 
trigonometric functions of 0, 30, 45, 60, and 90. Now we 
are able to consider the values of the trigonometric functions 
of related angles in other quadrants. 



56 



GENERAL DEFINITIONS 



[CHAP. Ill 



For example, to find the trigonometric functions of 240, draw 
the line OP (Fig. 12) so that angle XOP is 240. Therefore 

angle COP = 240 - 180 = 60. Take 
the distance OP as 2 units, draw PC 
perpendicular to the x-axis, and com- 
pute OC = - 1 and CP = - \/3. From 
the triangle OPC we read 

sin 240 = -\/3/2, 
cos 240 = -1/2, 
tan 240 = \/3, 
esc 240 = -2/V3, 
sec 240 = -2/1, 
cot 240 = l/\/3. 




-V3 



Fia. 12. 




-1 



-1 



,180 







(a) 



(6) 
Fio. 13. 




To illustrate the procedure further, we devise Figs. 13(a), 13(6), 
and 13 (c) and from them read the values tabulated below. 



TABLE A 



Angle 


sin 


cos 


tan 


cot 


sec 


CSC 


-45 


-VV2 


l/\/2 


-1 


-1 


V2 


-V2 


180 





-1 





oo 


-1 


00 

-1 


270 


-1 





00 





00 



To find the trigonometric functions of a special angle, the 
student should draw the angle, form a right triangle by dropping a 
perpendicular from a point on the terminal ray to the x-axis, write 
appropriate numbers on the sides of the right triangle, and read the 
values of the functions from the figure. 



22] VALUES OF TRIGONOMETRIC FUNCTIONS 57 

EXERCISES 

1. Draw a figure similar to Fig. 12 but designed for an angle of 
210. From this figure read the values of the trigonometric functions of 
210. 

2. Make a tabular form, similar to that of Table A above, containing 
a blank space for each of the values of the six trigonometric functions of 
0, 60, 90, 120, 135, -135, 270, -60, 315. Then fill in the blank 
spaces of the form from figures prepared for the purpose. 

3. Find two positive angles A less than 360 for which 

(a) sin A = |. (d) tan A = ~i\/3- 

(6) sin A = . (e) cos A = l/\/2- 

(c) tan A = i\/3. (/) sec A = - V2. 

4. Find all positive angles less than 360 for which 

(a) sin A 1. (d) cos A = 0. (g) cot A = 0. 

(6) cos A = 1. (e) sin A = 0. (h) tan A = oo. 

(c) tan A = 0. (/) esc A = -1. (i) cot A = oo. 

5. Find the values of the trigonometric functions of (a) 165; (6) 285; 
(c) 245; (d) 205; (e) 105. 

Hint. Use the table in 6. 

6. Evaluate 4\/3 tan 150 + 3 sin 90 tan 225 - 6 sin 330 + 
cos 270. 

7. Evaluate (a) sin 60 - 2 sin 330; (6) 2 sin 45 - sin 690; (c) 3 
cos 60 - cos 180; (d) 3 sin 690 - sin 90. 

8. Evaluate 4 sin 90 sin 330 sin 180 + (l/\/3) tan 240. 

9. Show that sin 120 = sin 180 cos 60 - cos 180 sin 60. 

10. Show that 

_ tan 240 - tan 30 
tan zw - ! + tan 24() o tan 30 o- 

11. Show that 

cos 120 cos 210 - sin 120 sin 210 



cot 330 
12. Verify that 



x . 3 tan - tan 3 
tan ,30=-! 3t - a -rr 



for each of the following values of 0: (a) = 45; (6) = 135; (c) 
= 120. 

13. Verify that sin 40 = 4 sin cos 0(cos 2 sin 2 0) for each of the 
following values of 0: (a) = 30; (6) = 120; (c) = 210. 



58 



GENERAL DEFINITIONS 



[CHAP. Ill 



14. Verify that sin (A + B) = sin A cos B + cos A sin B for (a) 



210 



30; (6) A = 135, 



225. 



16. Verify that cos (A + B) = cos A cos B sin A sin /? for (a) 
A = 120, = 210; (6) A = 315, B = 135. 
16. Evaluate: 



(a) 



cos 150 tan 300 
cot 225 + sin (-30)" 

sec 2 135 

cos (-240) - 2 sin" 2 10 



tan 3 315 

(c) 2 sin 2 240 + cos 180 



(d) 



sin 90 - 3 cot 495 
cos 510 esc (-00)' 



23. Fundamental identities. The fundamental identities (1), 
(2), (3), and (9) of Chap. II are true for all angles. The argu- 
ments used in Chap. II to prove (1), (2), and (9) for acute angles 
may be extended to apply to angles of any magnitude, provided 
no angles are considered for which any function involved is 
undefined; this may be done by replacing a by x, b by y, and 
c by r in those arguments. That the relations (3) of 11 are true 
also for all values of an angle A will be shown in Chap. V. Since 
only permissible algebraic operations and the identities just 
referred to were used in the verifications of Chap. II, all these 
verifications apply whether the angle is acute or not. 

24. Expressing a trigonometric function of any angle as a 
function of an acute angle. When the trigonometric functions 




FIG. 14. 



of an angle of any magnitude are read from a figure, they are 
always read from a right triangle, that is, from an acute-angled 
triangle. Hence it is always possible to express any one of the 



24] EXPRESSING A TRIGONOMETRIC FUNCTION 59 

six trigonometric functions of an angle as plus or minus a trigo- 
nometric function of a positive angle less than 90; in fact, they 
can be expressed as functions of an angle no greater than 45. 

Consider, for example, the problem of expressing the six 
trigonometric functions of 230 in terms of trigonometric func- 
tions of angles less than 90. 

In Fig. 14 angle XOP represents 230. OP - r, and line PR 
is drawn perpendicular to the x-axis. The length of OR is a, 
that of RP is 6, and the coordinates of P are x = a, and y = b 
as indicated. PO is prolonged into the first quadrant to Pi so 
that OPi = OP = r, and R\Pi is perpendicular to the x-axis. 
Therefore triangle* ORiPi is congruent to triangle ORP and PI 
is the point (a, b). Hence, using the definitions (1), we have 



But from triangle R } OP } , = sin 50. Hence 



sin 230 = - = - sin 50. 

Similarly, from Fig. 14 we obtain 

c^o^o 
cos 230 



^o^o -r(ofP) a /a\ 

30 = -- --- - = ----- = I - ) = cos 50. 

r r \r/ ' 



tan 230 = = = = tan 50. 

x(ofP) a a 

Continuing the same line of reasoning, we get 
cot 230 = r = cot 50, 

sec 230 = ~ = - sec 50, 

esc 230 = ^V = - esc 50. 
o 

Since for acute angles B 

/n(0) = co/ w (90 - 0) 
[see (3) 11], we have 

sin 230 = - sin 50 = - cos 40, 
cos 230 = - cos 50 = - sin 40, etc. 



60 



GENERAL DEFINITIONS 



[CHAP. Ill 



Hence the functions of 230 can be expressed as functions of 40, 
an angle less than 45. 

Similarly, to express the functions of 20 in terms of func- 
tions of 20, construct Fig. 15, and from it obtain 



sin (-20) = = - sin 20, 



cos (-20) = - = cos 20, 
tan (-20) = = - tan 20, 



esc (-20) = - esc 20, 
sec (-20) = sec 20, 
cot (-20) = - cot 20, 




It was pointed out in 21 that the values of the six trigono- 
metric functions of n 360 + A 
are respectively identical with 
those of A, provided n is any 
integer, positive or negative. 
Hence, to deal with -380, first 
add 360 to obtain -20, and 
then operate with 20 as above. 
FIQ 15 To deal with 950, first subtract 

720 = 2 X 360 to obtain 230, 
and then operate with 230 as above. 

EXERCISES 

PI 

1. In Fig. 16, OP = OP t . Use 
it to express the six trigonometric 
functions of 140 in terms of func- 
X tionsof40. 




\ 



2. Use Fig. 17 to express the trigo- 
nometric functions of 325 in terms of 
functions of 35. 



FIQ. 17. 



25] 



FUNCTIONS OF AND 180 



61 



3. Express the trigonometric functions of each of the following angles 
in terms of functions of an acute angle: 



(a) 243. 

(6) 326. 

(c) 198. 

(d) 170. 

(e) 310. 



(/) 155. 

(g) 350. 

(h) 470. 

(i) 545. 

(j) 730. 



(k) -200. 

(I) 99. 

(m) 2CO. 

(n) 130. 

(o) 925. 




R 



**X 



25. Functions of + and 180 + 6 in terms of functions of 0. 

The process used in 24 may, be used to get general formulas 
to be used in expressing functions of any angles in terms of func- 
tions of acute angles. Although the formulas will be derived 
under the assumption that is an acute angle, it will be proved 
later that they apply to the case when represents any angle. 

In Fig. 18 angle XOP is 
180 minus any acute angle 
0. P is any point different 
from O on ray OP, its coor- 
dinates are x = a, y = &, 
and it is distant r from the 
origin. PR is drawn per- 
pendicular to the x-axis, and 
triangle OP\R\ is drawn congruent to triangle OPR as indicated. 
Referring to Fig. 18, we find 

sin (180 - 0) = y 

and b/r in triangle OP^Ri is sin 0. Therefore 

sin (180 - 0) = sin 0. (2) 

Similarly 

cos (180 - 0) = - = -(-) = - cos 0, 



(3) 



tan (180 - 0) = ~ = -I ~] = - tan 0, 



cot (180 - 0) = 
sec (180 - 0) = 



a 

"F 

r 

a 



sec 



esc (180 - 0) = = esc 0. 



62 



GENERAL DEFINITIONS 



[CHAP. Ill 



In Fig. 19 angle XOP is equal to 180 + 0, where is an acute 
angle. The coordinates of P are x = a, y = 6, and the 




FIG. 19. 



FIG. 20. 



congruent triangles OPR and OP\Ri have been constructed as 
indicated. Referring to Fig. 19, we find 

sin (180 + 0) = = - sin 0, 
cos (180 + 0) = = - cos 0, 



tan (180 + 0) - = tan 0, 
cot (180 + 0) = ^| = cot 0, 
sec (180 + 0) = -^- = - sec 0, 
esc (180 + 0) = - = - esc 0. 



Similarly, from Fig. 20, we get 

sin (0) = = sin 0, esc ( 0) = =- = esc 0, 
r o 



(4) 



sec ( 0) = - = sec 0, 



cos ( 0) = - = cos 0, 

tan ( 0) = - = tan 0, cot ( 0) = r = cot tf. 

Considering formulas (2), (3), (4), and (5), we may write 

/n(180 0) = /n(0), /n(0) = /n(0), (6) 



(5) 



where fn refers to any one of the six symbols sin, cos, tan, etc., 
and the plus or minus sign in the right-hand member is to be 



25] FUNCTIONS OF AND 180 e 63 

used according as the left-hand member is a positive quantity 
or a negative quantity. 

Since any integral multiple of 360 may be added to an angle, 
equations (6) could be replaced by 

fn(klSO 0) = fnO (7) 

where k is an integer and the plus or minus sign in the right-hand 
member is to be used according as fn(k!8Q 6) is positive or 
negative. 

Example. For each of the following expressions write an 
equivalent expression involving only an acute angle: 

(a) cos 138, (6) tan 295, (c) sin 235. 

Solution, (a) cos 138 = cos (180 - 42) = - cos 42. The 
minus sign was chosen in the right-hand member because cos 138 
is negative. 

(6) Similarly tan 295 = tan (2 X 180 - 65) = - tan 66. 
The minus sign was chosen in the right-hand member because 
tan 295 is a negative quantity. 

(c) sin 235 = sin (180 + 55) = - sin 65. 

EXERCISES 

1. Use the method of this article to express the trigonometric functions 
of the following angles in terms of trigonometric functions of angles 
less than 90; (a) 265; (b) 275; (c) 125. 

2. For each of the following expressions use the method of this article 
to write an equivalent one in terms of an angle no greater than 45: 
sin 85, tan 338, sec 247, cos 197, cot 130, esc 500, sin 640, 
cos 1280, tan 2220. 

3. Express as trigonometric functions of B each of the following: 

(a) sin (360 - 0). (e) esc (2 X 180* + 0). 

(b) cos (720 - 20). (/) sin (360 - 26). 

(c) tan (180 - 0). (g) cot (30 X 90 + 0). 

(d) sec (540 - 6). (h) cos (0 - 360). 

4. Using trigonometric functions and positive angles less than 360, 
find three expressions equal to 

(a) sin 20. (e) sec 132. (i) cot 550. 

(6) cos 50. < (/) cot 247. (/) COB 635. 

(c) tan 75. (g) sin 328. (k) sin 740. 

(d) esc 87. (h) tan 432. 



64 GENERAL DEFINITIONS [CHAP. Ill 

5. Prove that sin 20 = sin 160 = cos 290 =- - sin 340. 

6. Simplify: 

+ cos 86 cos 94. 

tan 70 - sec 50 cos 130. 




cos ~r + tan (3^0 + 0) - sec (180 + 6) = sec 0. 



cot (180 + A) sin (360 - A) 
(b) cot (180 - A) - cos (360- A) = tan (72 + A) - L 

8. Prove that 

cos (90 + A) cos (270 - A) - sin (180 - A) sin (360 - A) 

= 2 sin 2 A. 

26. MISCELLANEOUS EXERCISES 

1. The tangent of a certain angle is , and its cosine is 3/\/l3. 
Find all the other trigonometric functions of this angle. 

2. Find all the trigonometric functions of a third-quadrant angle 
whose sine is f . 

3. Find two positive angles A less than 360 for which 

(a) sin A = ^ (c) cot A = l/\/2, (e) esc A = 2, 

(b) tan A = \/3, (d) sec A = V2> (/) cos A = . 

4. For each of the following expressions write an equivalent one 
in terms of an angle less than 90: 

(a) sin 105. (c) sec 340. (e) esc 290. 

(b) cos 170. (d) cot 242. (/) tan 184. 

5. For each of the following expressions write an equivalent one in 
terms of an angle no greater than 45: 

(a) sin 170. (c) cot 285. (e) sec 100. 

(6) cos 195. (d) tan 330. (/) esc 265. 

6. Find in radical form the value of each of the following: 

(a) cot 120. (c) sin 240. (e) sec 225. 

(6) cos 210. (d) esc 135. (/) tan 600. 

7. Evaluate: , 

sin 330 cos 135 cot 240 cos 150 
tan 225 cos 180 + sec 300 sinW ' 



26] MISCELLANEOUS EXERCISES 66 

8. Evaluate: 

esc 2 300 sin 60 tan 150 + sec 2 210 cot 240 cos 2 30. 

9. Simplify: 

cos 255 sec 75 sin 100 cos 260. 

10. Prove that 

sin 420 cos 390 + cos (-300) sin (-330) = 1. 

11. Prove that 

cos 570 sin 510 - sin 330 cos 390 = 0. 

12. Prove that 

tan y + tan (-a:) - tan (180 - x) = tan y. 

13. Prove that 

-"fi tan (90 + y} + csc2 (270 - y) - l + sec2 * 

14. Evaluate 4v/3 tan 330 + 3 sin 270 cos 90 - 6 sin (-30). 
16. Find in simple radical form the value of 

csc 225 sec 330 cos 690 + tan 240 sin 600 
cot 330 sin 240 - cos 210 cot 120 sin 270' 

16. Show that 

sin 240 = sin (-90) sin 120 - cos 270 cos (-60). 

17. Verify that sin 240 = 2 sin 120 cos 840. 

18. Verify that 

cos 255 = sin 45 sin 30 - cos 45 cos 30. 

19. Verify that sin 195 = sin 135 cos 60 + cos' 135 sin 60. 

20. Verify that sin (A + B) sin A cos B + cos A sin B for (a) 
A = 330, B = 60; (b) A - 135, B = 315. 

21. Verify that cos (A + B) = cos A cos B sin A sin B fpr (a) 
A = 30, B = 60; (6) A = 240, B = 330. 

22. Verify that 

4. /A m tan A - tan B 
tan < A - *> - 1+ tan A tan* 

for (a) A = 240, B = 120; (6) A = 315, B = 225. 



GENERAL DEFINITIONS 



[CHAP. Ill 



23. Verify that 

cos 3 A - cos 2 A cos A sin 2 A sin A, 
sin 3A = sin 2A cos A + cos 2A sin A, 

for (a) A = 60; (6) A = 135; (c) A = 600. 

24. Verify that 

tan 2 x esc 2 x cot 2 x sin 2 x = 1 

{a) x = 240, (6) x = 300, (c) x = 480. 
Verify that 

sin x + 1 cos x 

_ = t a n ^ + sec x 

sin x 1 + cos x 

for (a) x = 210, (6) x = 225, (c) x = 315, (d) x = 330. 

26. Verify that 

esc 2A = cot A cot 2A 

for (a) A = 120, (6) A = 210, (c) A = 225. 

27. Verify that 

sin (2x + y) + sin (2x y) 

- - . - =4 cos x cos y 

sin x 

for (a) x = 120, y = 60; (6) x = 150, y = 120. 



CHAPTER IV 
THE RIGHT TRIANGLE 

27. Introduction. In the study of the first chapter we solved 
a number of right triangles. Although the process in this chapter 
will be essentially the same as that used before, the treatment 
given here will be more thorough and complete. All cases will 
be considered, more complicated figures will be solved, and in 
some of the problems the computation will be carried out by 
means of logarithms. For this purpose tables that are more 
complete and accurate will be used. In practice, logarithms are 
employed when considerable accuracy is desired ; but when three- 
figure accuracy is sufficient the slide rule may be used. Triangles 
and rectilinear figures can be solved by means of the slide rule 
in a small fraction of the time required by logarithmic computa- 
tion; and even when extreme accuracy is desired, the slide-rule 
results serve as a rough check. 

28. Accuracy. Suppose a man knows that his house is longer 
than 31.5 ft. but shorter than 32.5 ft. How can he express the 
length of his house on the basis of this meager knowledge? If 
he should tell an engineer that his house was 32 ft. long, the 
engineer would be justified in thinking that the length was 
correct to the nearest foot. Hence he might argue as follows: 
The house is more than 31.5 ft. long; otherwise 31 ft. would be 
a closer approximation than 32 ft. Also, the house is shorter 
than 32.5 ft.; otherwise 33 ft. would be a better approximation. 
Similarly, if a man gave 32.3 ft. as the length of his house, an 
engineer would conclude that it was longer than 32.25 ft. but 
shorter than 32.35 ft. Evidently the error in this case would not 
be greater than y^ (= ^Q) ft., or 0.6 in. The first length, 
32 ft., would be spoken of as accurate to two significant figures, 
the second length, 32.3 ft., as accurate to three significant figures. 
A number is rounded off (or is accurate) to k significant figures 
when it is expressed, as nearly as possible, by means of a first 

67 



68 THE RIGHT TRIANGLE [CHAP. IV 

digit different from zero, k 1 digits immediately following the 
first, and enough zeros to place the decimal point. Thus 
0.000512 ft., 318000 in., 0.308 mile, all represent data accurate 
to three significant figures. Note that neither the four zeros 
in 0.000512 nor the three zeros in 318000 are significant, since 
they serve merely to place the decimal point. The numbers 
27862, 0.3996, and 38.85 when rounded off to three figures would 
be 27900, 0.400, 38.8, respectively. 38.85 might have been 
rounded off to 38.9; we chose 38.8 because many computers take 
the even digit when there is a choice. 

Results got by using a 10-in. slide rule are generally con- 
sidered accurate to three significant figures, although one cannot 
always be sure of the last figure. With data accurate to four 
figures four-place logarithm tables are used, with data accurate 
to five figures, five-place tables are used, etc. The result of 
computing 0.0038761 V4.8724 would be written 0.00856 if 
computed with a 10-in. slide rule, 0.008556 if computed with a 
four-place logarithm table, and 0.0085560 if computed with a 
five-place table or a more accurate one. 

EXERCISES 

1. Round off each of the following numbers to three figures, 
(a) 6.7245, (b) 984.55, (c) 69349, (d) 4935. 

2. A careless engineer gave the height of a flagpole as 48.672 ft. 
However, the measurements were made so poorly that his result might 
have been 2 in. in error. What height should have been given? 

29. Tables of natural trigonometric functions. By means of 
advanced mathematics the values of the trigonometric functions 
have been computed for a large number of angles. On page 69 
is listed the values, accurate to three figures, of the trigonometric 
functions for each degree from to 90. 

The value of a function of an angle between and 45 will be 
found in the row with the number of degrees in the angle and 
in the column headed by the name of the function. If the angle 
lies between 45 and 90, its value will be found in the row with 
the number of degrees in the angle and in the column having the 
name of the function at its foot. 

If the angle is not an exact number of degrees, the value of a 
function of the angle may be found by interpolation. For 



29] TABLES OF NATURAL TRIGONOMETRIC FUNCTIONS 69 



NUMERICAL VALUES OF THE TRIGONOMETRIC FUNCTIONS 



Degrees 


sin 


CSC 


tan 


cot 


cos 


sec 







000 


00 


0.000 


oo 


1.000 


1.000 





1 


0.017 


57.299 


0.017 


57.290 


1 000 


1 000 


89 


2 


035 


28 654 


0.035 


28.636 


0.999 


1 001 


88 


3 


052 


19 107 


052 


19.081 


0.999 


1 001 


87 


4 


070 


14.336 


0.070 


14.301 


0.998 


1.002 


86 


5 


087 


11.474 


0.087 


11.430 


0.996 


1.004 


85 


6 


105 


9 567 


105 


9.514 


0.995 


1.006 


84 


7 


0.122 


8 206 


123 


8 144 


0.993 


1 008 


83 


8 


139 


7 185 


0.141 


7.115 


0.990 


1.010 


82 


9 


0.156 


6.392 


0.158 


6.314 


0.988 


1.012 


81 


10 


174 


5 759 


0.176 


5.671 


0.985 


1.015 


80 


11 


191 


5.241 


0.194 


5 145 


982 


1 019 


79 


12 


208 


4.810 


213 


4 705 


978 


1 022 


78 


13 


225 


4 445 


231 


4 331 


974 


1 026 


77 


14 


0.242 


4 134 


249 


4.011 


970 


1 031 


76 


15 


259 


3 864 


268 


3.732 


0.966 


1 035 


75 


16 


276 


3 628 


287 


3 487 


961 


1 040 


74 


17 


292 


3 420 


306 


3 271 


956 


1 046 


73 


18 


309 


3 236 


325 


3 078 


951 


1 051 


72 


19 


326 


3.072 


0.344 


2.904 


0.946 


1 058 


71 


20 


342 


2 924 


0.364 


2 747 


940 


1 064 


70 


21 


358 


2 790 


384 


2 605 


934 


1 071 


69 


22 


375 


2 669 


404 


2 475 


927 


1 079 


68 


23 


391 


2 559 


424 


2 356 


921 


1 086 


67 


24 


407 


2 459 


445 


2.246 


914 


1 095 


66 


25 


423 


2.366 


466 


2 145 


906 


1 103 


65 


26 


438 


2 281 


488 


2 050 


899 


1.113 


64 


27 


454 


2 203 


510 


963 


891 


1.122 


63 


28 


409 


2 130 


532 


881 


0.883 


1.133 


62 


29 


485 


2.063 


0.554 


.804 


0.875 


1.143 


61 


30 


500 


2 000 


577 


.732 


866 


1 155 


60 


31 


515 


1.942 


601 


.664 


857 


1 167 


59 


32 


530 


1 887 


625 


600 


848 


1 179 


58 


33 


545 


1 836 


649 


.540 


839 


1 192 


57 


34 


. 559 


1.788 


0.675 


.483 


0.829* 


1.206 


56 


35 


574 


1 743 


700 


.428 


0.819 


1 221 


55 


36 


588 


1 701 


727 


376 


809 


1 236 


54 


37 


602 


1 662 


0.754 


327 


799 


1 252 


53 


38 


616 


1 624 


781 


.280 


788 


1 269 


52 


39 


0.629 


1 589 


810 


.235 


777 


1.287 


51 


40 


643 


1 556 


839 


192 


766 


1.305 


50 


41 


656 


1.524 


869 


150 


755 


1.325 


49 


42 


669 


1 494 


900 


.111 


743 


1.346 


48 


43 


682 


1 466 


933 


1.072 


0.731 


1.367 


47 


44 


695 


1 440 


0.966 


1.036 


719 


1.390 


46 


45 


707 


1 414 


1 000 


1.000 


707 


1.414 


45 




cos 


sec 


cot 


tan 


sin 


CSC 


Degrees 



70 THE RIGHT TRIANGLE [CHAP. IV 

example, to find sin 5724', take from the table the values of 
sin 57 and sin 58, and make the following form: 



60' 



( /sin 5700" = 0.839) ) 
< 24 \sin5724' = ? / > 
( sin 5800" = 0.848 ) 



For small changes in an angle, the increment of anglo is nearly 
proportional to the increment of its sine. Therefore 

f = g (nearly), or d = (f&)(9) = 4 (nearly). 

Adding 0.004 to 0.839, we obtain 

sin 5724' = 0.843. 

When the value of the function is given, a similar process 
enables us to find the angle. For example, if tan 6 = 0.734, to 
find 6 we use the table to get tan 36 = 0.727, tan 37 = 0.754, 
and then make the following form : 

( , (tan 36 = 0.727) ) 

60' < X (tan B = 0.734J > 27. 

( tan 37 = 0.754 ) 

x' 
As before, we write ^ = ^y, or x' = (^V)(60 X ) = 16' (nearly). 

Therefore x = 3616'. 

EXERCISES 

Find the value of each of the expressions numbered 1 to 6: 

1. sin 4240'. 4. cot 2035 / . 

2. cos 5423'. 6. sec 6220 / . 

3. tan 2210'. 6. esc 1618'. 

For each of the following equations, find an acute angle satisfying it: 

7. sin e := 0.672. 9. tan 6 = 1.630. 

8. cos 6 = 0.908. 10. cot e = 0.518. 

30. Solving right triangles. The sides and the angles of a 
rectilinear figure are called its parts. It is convenient, when no 
misunderstanding will result, to refer to a part of a figure or to its 
magnitude by the same name. When, for example, we speak 



30] 



SOLVING RIGHT TRIANGLES 



71 




of the hypotenuse of a right triangle we shall sometimes mean its 
longest side and sometimes the length of the longest side. The 
context will always indicate which meaning is intended. 

The conventional way of lettering a triangle is to assign, as 
was done in Fig. 1, the letters a, 6, c to the sideb and the letters 
A, B, C, respectively, to the angles 
opposite. 

When enough parts of a rectilinear 
figure are given to determine it, the 
process of finding the remaining parts 
is called " solving the figure." A right 
triangle is determined when a side and 
another part are given. The following italicized rule states the 
method to be used in solving a right triangle. 

Rule. To find an unknown part of a right triangle when a side 
and another part are given, (a) draw a representative figure, and write 
on each known part its value and on the unknown part a letter; (b) 
read from the figure a formula connecting the unknown part and 
the known parts; (c) solve for the unknown part, and compute its 
value. 

When all unknown parts of a triangle have been computed, the 
work may be checked by reading from the triangle an equation 
involving the computed parts, finding the B 

value of each member, and observing that 
these values differ very little if any. 



Example. Solve the right triangle in 
which a = 86.7 and 6 = 49.8. 

Solution. Construct Fig. 2 and from it 
obtain 



tan A = 



(o) 



a =86.7 



6=49.8 
FIG. 2. 



From the table of 29 and (a) find A = 608'. To get c, use 
Fig. 2 to obtain 



86.7 



= esc A, or c = 86.7 esc 608'. 



(b) 



Now replace esc 608' by 1.153, its value from the table of 29, 



72 THE RIGHT TRIANGLE [CHAP. IV 

to obtain 

c = 86.7 X 1.153 = 100.0. 

49 8 

To check, write ' = cos 608', or 49.8 = c cos 608'; replace 
c 

c by 100.0 and cos 608' by 0.498 to obtain 

49.8 = 100.0 X 0.498 = 49.8. 

EXERCISES 

Solve the following right triangles : 

1. a = 32, A = 4825'. 6. b = 67, B = 3215'. 

2. c = 46.1, B = 2914'. 6. c = 47.6, A = 6212'. 

3. c == 16.3, a = 25.1. 7. a = 41, b = 20. 

4. a *= 3.04, 6 = 2.51. 8. c = 37, A = 6950'. 

31. Definitions. T*he terms defined below will be used in the 
following list of problems and elsewhere in this book. 

The line of sight is a straight line connecting the eye of an 
observer with the object viewed. 

The angle of elevation at a point of an observed point 
B higher than is the angle that the straight line OB makes 
with the horizontal. 

The angle of depression at a point C of an observed point 
lower than C is the angle that the straight line CO makes with 
the horizontal. 

The angle subtended by a line BC at a point O is the angle 
formed by the rays OB and OC. 

For example, in the vertical plane OBC represented in Fig. 3, 
OB is the line of sight for an observer at viewing the point B, 
angle x is the angle of elevation of B at 0, angle y is the angle 
of depression of C at 0, and angle BOC is the angle subtended 
at by the line BC. 

The compass bearing of an object is the angle, measured clock- 
wise, that is, from north around toward or through east, between 
a horizontal line running north from an observer and a horizontal 
line connecting the observer with the object. The angle meas- 
ured clockwise in a horizontal plane from north to the direction 
of motion of an observer is known as his compass course. 



31] 



DEFINITIONS 



73 



Thus the bearing of point A for an observer at in Fig. 4 is 
130; the bearing of B is 330. A ship sailing from toward A 





Fw. 3. 

would have a compass course of 130. The direction to an 

object is often indicated by stating an initial direction, north (N.) 

or south 08.), then the angle in de- 

grees, minutes, and seconds, and 

finally a letter indicating whether 

the object is east (E.) or west (W.) 

of the observer. Thus the bearing 

of A in Fig. 4 might be given as S. 

50 E. and that of B as N. 30 W. 

When a ship sails a comparatively 
short distance from a point P to a 
point P' so as to cut at a constant 
angle a all meridians crossed by it, 
we use the words departure (Dcp), difference in latitude (DL), 
distance, and course in speaking of its trip. To understand the 
meaning of these words, consider the tri- 
angular figure PP'N (see Fig. 5) in which PP' N 
represents the path of the ship, PN repre- 
sents an arc of a meridian, and NP' repre- 
sents a "small" circle, all points of which 
have the same lattitudc. For practical 
purposes we consider this triangle as a plane 
right triangle and call distance NP f the 
departure, PN the difference in latitude, PP' 
the distance y and angle a the course. The 
course angle a. is measured from the north around through the 
east from to 360. 



Departure 




Fm 6 



74 



THE RIGHT TRIANGLE 



[CHAP. IV 



EXERCISES 

1. The master of a whaling vessel orders his mate to take a position 
500 yd. from his ship in a small boat, as shown in Fig. 6. The top of the 




FIG. G. 

whaling vessel's mast above the water line is 213 ft. Find what angle 
this height will subtend on the mate's sextant when he reaches his 
position. 

2. A ship moving due west at 15 mile per hour passes due north of a 
given point A, and 20 min. later it bears N. 382(>' W. from the given 
point. Find the distance of the ship from A at both times. 

3. A surveyor in a barn distant I mile from a railroad track observes 
that a train of cars on the track subtends 3540' at his position when one 




FKJ. 7. 

end of the train is directly opposite him. How long is the train (see 
Fig. 7)? 

4. From the top of a rock that rises vertically 80 ft. out of the water 
the angle of depression of a boat is found to be 35; find the distance 
of the boat from the foot of the rock. 

6. The shadow of a vertical cliff 1 13 ft. 
high just reaches a boat on the sea 93 ft. 
from its base; find the altitude of the sun. 
6. The rafters of a house make an angle 
of 4550' with the horizontal and are 16 ft. 
long from the top of the wall to the high- 
est point of the roof. Find the height h 
of the roof above the wall (see Fig. 8). 




FIG. 8. 



31] 



DEFINITIONS 



75 



7. The two stations A and B shown in Fig. 9 are 5200 ft. apart. 
When an airplane D was directly above A an observer at B found the 



5200' 1" 



FIG. 9. 

angle of elevation of the plane to be 4()20 / . Find the distance from the 
plane to station B. 

8. From a point 1420 it. above a trench, an observer in an airplane 
finds that the angle of depression of an enemy fort is 2350 ; . How far is 
the trench from the fort? 

9. If a ship sails on a course of 42 for 190 miles, what are the 
departure and difference in latitude? 

10. A ship asks bearings from two radio stations A and B. A 
reports the ship's bearing 82 (Navy Compass) and B reports 127. 
Station B is known to be 127 nautical miles from A on bearing 58 from 
A . Find the difference in latitude and departure of the ship from A. 

11. From a point A 175 ft. from the base of a lighthouse a yachtsman 
finds the angle of elevation of the top to be 2930', as shown in Fig. 10. 
Find the height of the lighthouse. 




FIG. 10. 



76 



THE RIGHT TRIANGLE 



[CHAP. IV 



12. From an observer's position 0, 8.5 ft. above the water (see 
Fig. 11), the angle of elevation of the top B of the sail was found to be 




FIG. 11. 

2830', and the angle of depression of the lowest point C was 2025'. 
Find the total height BC of the sailboat. 

13. From the top of a hill the angles of depression of two successive 
milestones on a straight level road leading to the hill are observed to be 
5 and 15. How high is the hill? 

32. Solution of the right triangle by slide rule.* A funda- 
mental law of trigonometry, called the law of sines, is especially 

adapted to slide-rule computation. It 
states that the ratio of the sine of any 
angle of a triangle to the opposite side is 
equal to the ratio of the sine of any 
a second angle to its opposite side; or, in 
symbols, 



B 




sin A sin B sin C 



(1) 



To prove this for a right triangle, use Fig. 12 to obtain 



- = sin A, 
c 



or 



or 



1 
c 
1 
c 



sin A 

a 
sin B 



(2) 
(3) 



* A good preparation for making the computations of this article and the 
next one may be obtained by studying 127, 128. 



32] SOLUTION OF THE RIGHT TRIANGLE 77 

Equating the values of 1/c in (2) and (3), we get (sin A)/a = 
(sin B)/b = 1/c, or replacing 1 by its equal, sin 90, 

sin A sin B sin 90 ,.. 



To solve the triangle of Fig. 13, substitute 35 for A, 387 for a, 
and 55 for B in (4) to obtain 

S sin 35 sin 55 sin 90 




D "387" ~~ ~~b ~' 

where the symbol S/Z) indicates that 
the angles in the numerator are to be 
set on the S scale of the slide rule, and 
the denominators on the D scale. 
Hence, in accordance with the proportion principle, 

push hairline to 387 on Z), 

draw 35 of S under the hairline, 

push hairline to 55 on S, 

at the hairline read b = 562 on D; 

push hairline to 90 on S, 

at hairline read c = 675 on D. 

The student should note that it is unnecessary to write the 
law of sines to solve a right triangle. Observing that, in accord- 
ance with the law of sines, each side and the angle opposite 
must be set opposite each other on the slide rule, he uses the 
following rule: 

Rule. To solve a right .triangle, except when the given parts 
are two legs, draw the triangle and write on each known part, 
including the 90 angle, its value, and then 

push the hairline to known side on Z), 
draw angle opposite on S under hairline, 
push hairline to any other known side on D; 
at the hairline read angle opposite on S, 
push hairline to any known angle on S, 
at the hairline read side opposite on D. 



78 



THE RIGHT TRIANGLE 



[CHAP. IV 



4. b = 200, 
A = 64. 


7. b = 47.7, 
= 6256'. 


6. c = 37.2, 

B = 612'. 


8. a = 0.624, 
c = 0.910. 


6. c = 11.2, 
A = 4330 / . 


9. o = 83.4, 
A = 727'. 



EXERCISES 

Solve the following right triangles by means of the slide rule. 



l.o = 60, 
c = 100. 

2. a = 50.6, 
A = 3840'. 

3. a = 729, 

B = 6850'. 



33. Slide-rule solution of a right triangle when two legs are 

known. When the two legs of a 
right triangle are known, the smaller 
acute angle may be found from its 
tangent, the other acute angle by 
a =3.14 subtracting the smaller one from 
90, and then the hypotenuse by 
using the law of sines. Thus, to 
solve the right triangle shown in 
Fig. 14, write 



6=4.40 
FIG. 14. 



or 



tan A 
lU4~ 



1 
4.40* 



Hence, in accordance with the proportion principle, 

set the index of C to 440 on D, 

push hairline to 3.14 on D, 

at the hairline read A = 3631' on T. 

Evidently angle B = 90 - A = 5429'. To find the hypote- 
nuse c, apply the setting based on the law of sines explained in 
32; this leads us to: 

push hairline to 3.14 on D, 

draw 3531' on S under the hairline, 

at the index of C read c = 5.40 on D. 

If one observes that the first of the three steps just indicated is 
unnecessary, since the hairline was already set to 3.14 on D 



35] TRIGONOMETRIC FUNCTIONS OF AN ANGLE 79 

when the angle A was found, he will see that the following rule 
applies: 

Rule. To solve a right triangle when two legs are known: 

To greater leg on D set proper index of slide, 
push hairline to smaller leg on D, 
at the hairline read smaller acute angle on T, 
draw this angle on 8 under the hairline, 
at index of slide read hypotenuse on D. 

EXERCISES 

Solve the following right triangles by means of the slide rule: 

1. a = 12.3, 4. a = 273, 7. a = 13.2, 
b = 20.2. 6 = 418. b = 13.2. 

2. a = 101, 6. a = 28, 8. a = 42, 
b = 116. b = 34. 6 = 71. 

3. a = 50, 6. a = 12, 9. a = 0.31, 
b = 23.3. 6 = 5. 6 = 4.8. 

34. Table of logarithms of trigonometric functions. When 
a high degree of accuracy is desired for the solution of a problem 
involving trigonometry, the computation should be done by 
means of logarithms. To facilitate the process, tables of log- 
arithms of the trigonometric functions have been prepared. The 
sample page printed in the next article is a page from such a 
table. The complete table gives, accurate to five decimal 
places, the logarithms of the six trigonometric functions for 
angles from to 45 at intervals of 1 min. It may be applied 
directly for all positive angles less than 180. Tabular differences 
of successive logarithms are given in the columns headed d 1'; 
they are used in the process of interpolation that is designed to 
take account of seconds of angle. 

35. To find the logarithms of a trigonometric function of an 
angle. The solution of the following example illustrates the 
method of finding the logarithm of a trigonometric function of a 
given angle. 

Example. Find log sin 3542'17". 



80 



THE RIGHT TRIANGLE 



[CHAP. IV 



35 



144 



f 


I sin 
0. 


d 

1' 

18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
17 
18 
18 
18 
18 
18 
17 
18 
18 
18 
17 
18 
18 
18 
17 
18 
18 
17 
18 
18 
17 
18 
18 
17 
18 
18 

tf 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
17 
18 


icsc 
10. 


{tan 
9. 


d 
1' 

27 
26 
27 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
26 
27 
27 
27 
2l> 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
2b 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
26 
27 
27 
26 
27 
26 
27 
27 
26 


tcot 
10. 


{sec 
10. 


d 

r 


f COS 

9. 


' 




,. 


27 


i*ro 
26 


portic 
18 


mall 
17 


tats 
10 


9 


8 




1 

2 
3 

4 


75859 
877 
895 
913 
931 


24141 
123 
105 
087 
069 


84523 
550 
576 
603 
630 


15477 
450 
424 
397 
370 


08664 
672 
681 
690 
699 


8 
9 
9 
9 
9 
9 
9 
8 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
N 
9 
9 
9 

y 

9 
9 
9 
9 
9 

y 

9 
9 
10 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
10 
9 
9 
9 
9 
9 
9 
10 
9 
9 
9 
9 


91336 
328 
319 
310 
301 


60 

59 
58 
57 
56 




1 
2 
3 
4 




1 
1 
2 




1 
1 
2 




1 
1 
1 





1 
1 
1 






1 







1 







1 


5 

6 
7 
8 
9 


949 
967 
985 
70003 
021 


051 
033 
015 
23997 
979 


657 
684 
711 
738 
764 


343 
316 
289 
262 
236 


708 
717 
726 
734 
743 


292 
283 
274 
266 
257 


55 

54 
53 
52 

51 


5 

6 

7 
8 
9 


2 
3 
3 

4 

4 


2 
3 
3 
3 
4 


2 
2 
2 

2 
3 


I 

2 
2 

2 
3 


1 

1 

1 
I 

2 


1 
1 

1 
I 
1 




10 

11 
12 
13 
14 


039 
057 
075 
093 
111 


96l 
943 
925 
907 
889 


791 
818 
845 
872 
899 


209 
182 
155 
128 
101 


752 
761 
770 
779 

788 


248 
239 
230 
221 
212 


50 

49 
48 
47 
46 


10 

11 
12 
13 
14 


4 
5 
5 
6 

6 


4 
5 
5 
6 
6 


3 

3 
4 
4 

4 


3 
3 
3 
4 
4 


2 
2 
2 

2 
2 


2 
2 
2 
3 
2 


2 
2 
2 


15 

16 
17 
18 
19 


129 
146 
164 
182 
200 


871 
854 
836 
818 
800 


925 
952 
979 
85006 
033 


075 
048 
021 
14994 
967 


797 
806 
815 
824 
833 


203 
194 
185 
176 
167 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 


15 

16 
17 
18 
19 


7 
7 

8 
8 
9 


6 
7 

7 
8 
8 
" 9 
9 
10 
10 
10 


4 
5 
5 

5 
6 


4 
5 
5 
5 

5 


2 
3 
3 
3 
3 


2 
2 
3 
3 
3 


2 

2 
2 
2 
3 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


218 
236 
253 
271 
289 


782 
764 
747 
729 
711 
693 
676 
658 
640 
622 


059 
086 
113 
140 
166 


941 
914 
887 
860 
834 
807 
780 
753 
727 
700 


842 
851 
859 
868 
877 


158 
149 
141 
132 
123 
114 
105 
096 
087 
078 


20 

21 
22 
23 
24 


9 

9 
10 

10 

11 

11 

12 
12 
13 
13 


6 

6 
7 
7 

7 
8 
8 
8 
8 
9 


6 
6 
6 
7 
7 
7 

7 
8 
8 
8 


3 
4 
4 
4 
4 
4 
4 
4 
5 
5 


3 

3 
3 
3 
4 
4 
4 
4 
4 
4 


3 
3 
3 

3 
3 


307 
324 
342 
360 
378 


193 
220 
217 
273 
300 


886 
895 
904 
913 
922 


35 

34 
33 
32 
31 




25 

26 
27 

28 
29 


11 

11 

12 
12 

13 


3 
3 
4 

4 
4 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


76395 
413 
431 
448 
466 


23605 
587 
569 
552 
534 


85327 
354 
380 
407 
434 
460 
487 
514 
540 
567 


14673 
646 
620 
593 
566 


08931 
940 
949 
958 
9G7 


91069 
060 
051 
042 
033 


30 

29 
28 
27 
26 


30 

31 
32 
33 
34 


14 
14 

14 
15 

15 


13 

13 
14 
14 
15 
15 
16 
16 
16 
17 


9 

9 
10 
10 

10 


8 
9 
9 
9 
10 


5 

5 
5 
6 
6 


4 
5 
5 
5 
5 


4 

4 
4 
4 
5 


484 
501 
519 
537 
554 


516 
499 
481 
463 
446 


540 
513 
486 
460 
433 


977 
986 
995 
09004 
013 


023 
014 
005 
90996 
987 


25 

24 
23 
22 
21 


35 

36 
37 
38 
39 


16 
16 
17 
17 

18 


10 

11 
11 

11 

12 


10 

10 
10 

11 
11 


6 
6 

6 
6 
6 


5 
5 
6 
6 
6 


5 
5 
5 
5 

5 


40 

41 
42 
43 
44 


572 
590 
607 
625 
642 


428 
410 
393 
375 
358 


594 
620 
647 
674 
700 


406 
380 
353 
326 
300 


022 
031 
040 
049 
058 


978 
969 
960 
951 
942 


20 

19 
18 
17 
16 


40 

41 
42 
43 
44 


18 

18 
19 
19 
20 


17 
18 
18 
19 
19 


12 

12 
13 
13 

13 


11 

12 
12 

12 
12 


7 
7 
7 
7 
7 


6 

6 
G 
6 
7 


5 
5 
6 
6 
6 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


660 
677 
695 
712 
730 


340 
323 
305 
288 
270 


727 
754 
780 
807 
834 


273 
246 
220 
193 
166 


067 
076 
085 
094 
104 


933 
924 
915 
906 
896 


15 

14 
13 
12 
11 


45 

46 
47 
48 
49 


20 
21 
21 
22 
22 


20 
20 
20 
21 
21 
22 
22 
23 
23 
23 


14 
14 
14 

14 
15 


13 
13 
13 
14 
14 


8 
8 
8 
8 
8 


7 
7 
7 

7 
7 


6 

6 
6 
6 
7 


747 
765 
782 
800 
817 


253 
235 
218 
200 
183 


860 
887 
913 
940 
967 


140 
113 
087 
060 
033 


113 
122 
131 
140 
149 


887 
878 
869 
860 
851 


10 

9 
8 
7 
6 


50 

51 
52 
53 
54 
~55~ 
56 
57 
58 
59 


22 
23 
23 
24 

24 


15 

15 
16 
16 

16 


14 
14 
15 
15 

15 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


7 
7 
7 

7 
7 


55 

56 
57 
58 
59 
00 


835 
852 
870 
887 
904 


165 
148 
130 
113 
096 


993 
86020 
046 
073 
100 


007 
13980 
954 
927 
900 


158 
168 
177 
186 
195 


842 
832 
823 
814 
805 


5 

4 
3 
2 

1 




25 
25 

26 
26 

27 


24 

24 
25 
25 

26 


16 
17 
17 

17 
18 


16 
16 

16 
16 
17 


9 
9 
10 
10 
10 


8 
8 
9 
9 
9 


7 
7 
8 
8 
8 


76922 


23078 


86126 


13874 


09204 


90796 


J 

/ 


60 


27 


26 


18 


17 


10 


9 


8 

nr 


f 


9. 

J COB 


d 
1' 


10. 

I sec 


9. 

I cot 


d 
1' 


10. 

2 tan 


10. 

/CSC 


d 

1' 


9. 

I sin 




27 


26 

Pro 


18 
portic 


17 

nail 


10 

>arts 


9 



125 C 



54 C 



THE ANGLE WHEN THE LOGARITHM IS GIVEN 81 

Solution. From the table we find the logarithms in the follow- 
ing form and then compute the differences exhibited. 

log sin 35t2'00" \ r , } = 9.76607 - 10 \ ) 

log sin 3542'17" j > 60" = x f y > d = 18 

log sin 3543'00" ) = 9.76625 - 10 ) 

The small changes in angle are nearly proportional to the corre- 
sponding changes in logarithm. Therefore 

is = 60' or y = ( l8 ) 66 = 5 ( noarl y)- 

and log sin 3542'17" = 9.76607 - 10 + 0.00005 = 9.76612 - 
10. 

To perform the interpolation by means of the proportional- 
parts column, read 9.76607 10 as the log sin 35 12'; near this 
entry in the column headed d 1' note the number 18, in the pro- 
portional parts column headed 18 and in the row with 17 of the 
column headed " road 5, and add 0.00005 to 9.76607 - 10 to 
obtain 9.76612 - 10. 

EXERCISES 

Find the value of the following: 

1. log sin 3946'17". 6. log sin 6447'51". 

2. log sin 5<)31'26". 7. log tan 2011'11". 

3. log cos 8121'43". 8. log esc 1617'18". 

4. log tan 2829'49". 9. log sec 8119'31". 

5. log cot 4916'21". 10. log cos 1219'14". 

36. To find the angle when the logarithm is- given. The solu- 
tion of the following example illustrates the method of finding an 
angle when the logarithm of a trigonometric function of the 
angle is given. 

Example. Find the acute angle B when log tan B = 0.14920. 

Solution. Observe that 0.14920 lies between the two entries 
0.14914 and 0.14941 on the sample page in the column with I tan 
written at its foot. Therefore write the logarithms in the 
following form and compute the differences exhibited: 



82 THE RIGHT TRIANGLE [CHAP. IV 

log tan 5439' ) ) = 0.14914 ) ) 

log tan B ) y > 60" = 0.14920 / > d = 27. 

Iogtan5440' ) =0.14941 ) 

The small changes in angle are nearly proportional to the 
small changes in the logarithm. Therefore 

I) =4' or y = < 6 ) if = 13 "> 

and 

B = 6439'13" (nearly). 

To get the correction y by the proportional parts table: find 
the tabular difference 27 between the entries 14914 and 14941 
of the tangent column; find the difference 14920 14914 = 0; 
opposite the bold-faced 6 in the proportional parts column headed 
27 read 13 in the seconds column. Whenever there is a choice 
between two or more entries, one of which is printed in bold face, 
always give preference to the bold-faced entry. 

EXERCISES 

Find the value of A in the following: 

1. log sin A = 9.31461 - 10. 6. log cos A = 9.21611 - 10. 

2. log tan A = 9.03141 - 10. 7. log tan A = 0.11161. 

3. log cot A = 0.01210. 8. log cot A = 9.86192 - 10. 

4. log sin A = 9.12867 - 10. 9. log sin A = 9.02218 - 10. 
6. log cos A = 9.92112 - 10. 10. log sec A = 0.21210. 

37. Solution of the right triangle by means of logarithms. To 
solve a right triangle by means of logarithms, proceed as indi- 
cated in 30, but do the computation with a table of logarithms. 
The solution of the following example will indicate a very con- 

venient form for the computation 
as well as the method of procedure. 




Example. Solve the right triangle 
in which c = 796.47, a = 267.53. 

Solution. Fig. 15 shows the given 
parts encircled. From it we obtain 



37] SOLUTION OF THE RIGHT TRIANGLE 83 

A a f \ 

sin A = -> (a) 

B = 90 - A, (6) 

- = cos A, or 6 = c cos .4, (c) 

c 

- = cot A, or b = a cot A. (Check formula) (rf) 

From (a), 

log sin A = log n + colog c. 
From (c), 

log; /; = ]ojr r -f- log COS A. 

From (d), 

log /> = log a + log cot A. 

The following forms contains all numbers used in the compu- 
lation, including the results. Note that every expression on any 
line refers to the first number in the line 



(<0 (r) (d) 



a - 207 53 
c = 79(> -17 



logn = 2 42737 
rologr = 7 09883 - 10 I log c = 2 90117 



log a -= o 42737 
log cot A =0 44780 



A - 1937'37" loRHiii^ = 9 521420 - l() [1<>K <'<> ^ = 97401 - 10 
6 = 75020 logh = 2.87518 lug 6 - 2 87517 

B - 90 A = 7022'23". 

EXERCISES 

vSolve the following right triangles: 

1. b = 14, 6. c = 672.34, 9. A = 4410'38", 
A = 35. A = 3516'25". c = 24.896. 

2. c = 6.275, 6. a = 645.32, 10. a = 3.2914, 
B = 1847'. b = 396.25. 6 -= 5.7842. 

3. c = 1200.7, 7. c = 98.245, It. a = 72.131, 

a = 885.6. a = 95.573. A = 7617 / 32". 

4. a = 8.67892, 8. 7? = 279'14", 12. r = 1672.1, 

6 = 2.4639. a = 35.421. U = 83 2rir'. 

13. A stay wire for a telephone pole is to he attached to the pole 18 ft. 
6 in. above the ground and to make an angle of 42 10' with the hori- 
zontal. Find the length of the stay wire, allowing 3 ft. to make 
attachment. 

14. If a ship sails a course of 19 for 201.85 miles, what is the 
departure? 



84 



THK RIGHT TRIANGLE 



[CHAP. IV 




16. An observer in an air- 
plane 2500 ft. above the sea 
sights a destroyer at an angle of 
depression of 5730', as shown 
in Fig. 10. Find the distance 
between the plane and the 
destroyer. 



FIG. 10. 

16. If a railroad track rises 30 ft. 4 in. in a horizontal distance of 
5280.7 ft., what is its angle of inclination with the horizontal? 

17. The area of a right triangle is 23.577 sq. ft., and one angle is 
5224'29". Find the length of the hypotenuse. 

18. A diagonal of a cube intersects a diagonal of one of its faces. 
Find the angle between these diagonals. 

19. A marble * in. in diameter subtends an angle of 215'30" at the 
eye of an observer. How far is it from the observer? 

20. If two straight stretches of railway were extended they would 
meet at a point making an angle of 4(>18' with each other. These two 
stretches are to be connected by means of a circular arc of radius 4500 ft. 
Find the distance from the point of tangency to the point of intersection 
of the straight stretches. 

21. A rectangular bin is 42 in. long and 30 in. wide. What angles 
does a vertical, diagonal partition make with the sides of the bin? 

22. In building a suspension bridge a straight cable is run from the 
top of a pier to a point 852 ft. 7 in. from its foot. If from this point the 
angle of elevation of the top of the pier is 276', what length of cable is 
required? 

23. In a level field a tunnel was dug into the earth at an angle of 
1920' with the horizontal. At a point in the field 285 ft. from the 
entrance of the tunnel an engineer dug a vertical shaft to meet the 
tunnel. Find the depth of this shaft. 

24. Assuming that the earth is a sphere of radius 3958.5 miles, how 
far is a point in latitude 4140' from the earth's axis? 

26. On a 2 per cent railroad grade, that is, a rise of 2 ft. in each 
100 ft. measured horizontally, what is the angle at which the rails are 



38] SOU 'T ION OF RECTILINEAR FIGURES 85 

inclined to the horizontal? How far must one move along the rails 
to be 162 ft. higher than at the starting point? 

26. Find the radii of the inscribed and circumscribed circles of a 
regular octagon whose side is C.2538. 

27. At a point A due west of the Washington Monument, which is 
55,5 ft. high, the angle of elevation of its to]) was observed to be 5122.9'. 
Find the angle of elevation of the monument at another point 7>, 200 ft. 
west of A , assuming that the points A and B and the base of the monu- 
ment are in the same horizontal plane. 

38. Solution of rectilinear figures. The process of expressing 
line segments in terms of specified parts of a rectilinear figure 
was employed in Chap. IT. To compute the length of a line 
segment, or the magnitude of an angle forming part of a recti- 
linear figure, use the process of Chap. II to find an expression 
for the desired part, and Mien evaluate this expression. 

An expression is convenient for logarithmic computation if 
its evaluation involves only multiplications and divisions. To 
obtain such an expression for an unknown length in a rectilinear 
figure, one generally drops perpendiculars in such a way as to 
form a chain of right triangles, each of which has a side in com- 
mon with the next one in the chain. The first triangle has a side 
of known length, and the last one has as a side the length to be 
found. The; following example will illustrate the procedure. 

Example. A surveyor on a mountain peak observes below 
him two ships lying at anchor 1 mile apart and in the same 




FIG. 17. 

vertical plane with his position. He, finds the angles of depres- 
sion of the ships to be 18 and 10, respectively. How high does 
the peak rise above the water? 

Solution. In Fig. 17, // represents the position of the sur- 
veyor, M and N represent the respective positions of the ships, 



86 THE RIGHT TRIANGLE [CHAP. IV 

and the angles marked 10 and 18 represent the angles of 
depression. Draw NP perpendicular to MH, and denote the 
length of NP by x and that of NH by y. From triangle MNP, 

^ = sin 10, or x = 5280 sin 10. (a) 

uZoO 

From triangle NPH, 

- = csc 8, or y = x csc 8. (6) 

From triangle LNH, 

- = sin 18, or h = y sin 18. (c) 

y 

Substituting the value of y from (6) and x from (a) in (c), we 
obtain 

h = y sin 18 = x csc 8 sin 18 = 5280 sin 10 csc 8 sin 18. 
The following form shows the computation : 

log 5280 =3.72263 

log sin 10 - 9.23967 - 10 

log csc 8 = colog sin 8 = 85644 
loft sin 18 = 9 . 48998 

h = 2036.7 log Ji ^3 30872 

Too much accuracy is indicated by this answer for ordinary 
measurements. The surveyor might be justified in writing 
2.0 X 10 3 ft. or even 2040 ft. as the height of the peak. 

EXERCISES 

1. Two points A and B 80 ft. 
apart lie on the same side of a tower 
and in a horizontal line through its 
foot. If the angle of elevation of the 

lf top of the tower at A is 21 and at B is 

^ * E 46, find the height of the tower (see 

FlG ' 18 ' Fig. 18). 

Q\Two points A and B 180 ft. apart lie on the same side of a tower 
on ahill and in a horizontal line passing directly under the tower. The 
angles of elevation of the top and bottom of the tower viewed from B 
are 42 and 34, respectively, and at A the angle of elevation of the 
bottom is 10. Find the height of the tower. 

Hint. Draw Fig. 19, compute angle ACB = 24, angle EBC = 8, 
and note that angle EOF = 42. Find in order pi, BC, p 2 , and h. 




38] 



SOLUTION OF RECTILINEAR FIGURES 



87 



180' 



3. (a) Express BC, DE, and CE 
in terms of m and A (see Fig. 20). 

(6) (iiven m = 1.96 in. and tan 
A = 0.482, find BC, DE, and CE. 



4. (a) Express all line seg- 
ments of Fig. 21 in terms of a 
and <p. 

(b) Given a = 34.368, tan <p = 
0.30517; use logs to find the 
length of MN. 



5. Find the length of diameter 
AOB 9 the length of arc ADB, and the 
area of the semicircle shown in Fig. 
22. 




FIG. 22. 



88 



THE RIGHT TRIANGLE 



[CHAP. IV 




6. Given the angles a, ^>, 0, and 
the distance AB = m in Fig. 23; find 
formulas for x and y. 



7. Given AB parallel to 
CD, in Fig. 24, find the area 
of the figure ABDC. 



FIG. 24. 



8. A mountain peak C is 4135 ft. above sea level, and from C the 
angle of elevation of a second peak B is 5. An aviator at A directly 




FIG. 25. 

over peak C finds that angle CAB is 4350' when his altimeter shows 
that he is 8460 ft. above sea level. Find the height of peak B (see 
Fig. 25). 

(JA tower and a monument 
stand on a level plain' (see Fig. 26). 
The angles of depression of the top 
and bottom of the monument viewed 
from the top of the tower are 13 and 
31, respectively; the height of the 
tower is 145 ft. Find the height of 
the monument. 




FIG. 26. 



39] 



MISCELLANEOUS EXERCISES 



89 




10. As the altitude of the sun decreased from 6346' to 5035', the 
length of the shadow of a tower increased 89.65 ft. Find the height 
of the tower. 

11. Figure 27 represents a 600-ft. radio 
tower. AC and AD are two cables in the 
same vertical plane anchored at two points 
C and D on a level with the base of the 
tower. The angles made by the cables with 
the horizontal are 44 and 58 as indicated. 
Find the lengths of the cables and the dis- 
tance between their anchor points. 

12. A building and a tower stand on the same horizontal plane, the 
tower being 120 ft. high. From the top of the tower the angles of 
depression of the top and bottom of the building are 2213.8' and 
44 18.9', respectively. Find the height of the building. 

13. A line AB along one bank of a stream is 315 ft. long, and C is a 
point on the opposite bank. The angle BAG is 6630', and the angle 
ABC is 5445'. Find the width of the stream. 

14. From a ship two lighthouses bear N. 40 E. After the ship sails 
at 15 knots on a course of 135 for 1 hr. 20 min., the lighthouses bear 
10 and 345. 

(a) Find the distance between the lighthouses. 

(b) Find the distance from the ship in the latter position to the 
further lighthouse. 

39. MISCELLANEOUS EXERCISES 

Solve the following right triangles : 



l.o = 104, 
c = 185. 

2. c = 625, 
A = 44. 



3. 6 = 47.78, 
B = 3922'. 

4. a = 49967, 

B = 6243'34". 



6. c = 5.8902, 
7* = 678 / 20 // . 

6. a = 4.0007, 
6 = 7.9234. 



7. Two straight roads cross at an angle of 5236', and there is a 
town on one road 6520 yd. from the crossing. How far is this town 
from a point on the other road 2528 yd. from the crossing? (Give 
two answers.) 

8. The^ Pennsylvania Railroad _ n 

found it necessary, owing to land 

slides upon the roadbed, to reduce the 

angle of inclination of one bank of a 

certain railway cut near Pittsburgh, 

Pa., from an original angle of 45 to a 

new angle of 30, as shown in Fig. 28. Crosg gcction 

The bank as it originally stood was 200 Fro. 28, 




45 



90 



THE RIGHT TRIANGLE 



[CHAP. IV 



ft. long and had a slant length of 60 ft. Find the amount of the earth 
removed if the top level of the bank remained unchanged. 



9. A slide in a machine is to run on rolling 
balls. The balls run in grooves with straight 
sides as shown in Fig. 29. The angle of the 
upper (moving) groove is 120, and that of 
the lower (fixed) groove is 90. What size of 
balls should be used? 




FIG. 29. 



10. A searchlight situated on a straight coast has a range of 43 miles. 
A ship sails on a line parallel to the coast and 15 miles from it. What 
is the distance covered by the ship while it remains within range of the 
light? What angle is subtended at the light by a line connecting the 
extreme positions of the ship? 

11. A man in a balloon observes that the straight line connecting 
the bases of two towers, which are 1 mile apart on a horizontal plane, 
subtends an angle of 70. If he is exactly above the middle point of 
this line, find the height of the balloon. 




12. Find the number of square feet of pavement 
required for the shaded portion of the streets 
shown in Fig. 30, all the streets being 50 ft. wide. 



Fio. 30. 



13. A flagstaff 25 ft. high stands on the top of a house. From a 
point on the plain on which the house stands, the angles of elevation 
of the top and the bottom of the flagstaff are observed to be 60 and 45, 
respectively. Find the height of the house. 

14. From a point A 10 ft. above the water, the angle of elevation of 
the top of a lighthouse is 46, ancLrthe angle of depression of its image 
in the water is 50. Find the height h of the lighthouse and its hori- 
zontal distance from the observer (see Fig. 31). 



MISCELLANEOUS EXERCISES 



91 




g 42 ' 



FIG. 31. 

15. The pilot in an airplane observes the angle of depression of a 
light directly below his line of flight to be 30. A minute later its angle 
of depression is 45. If he is flying horizontally in a straight course at 
the rate of 150 miles per hour, find (a) the altitude at which he is flying; 
(b) his distance from the light at the first point of observation. 

16. From the top of a building the angle of depression of a point 
in the same horizontal plane with the base of the building is observed 
to be 47 13'. What will be the angle of depression of the same point 
when viewed from a position half way up the building? 

Ap The captive balloon C shown in Fig. 32 is connected to a ground 
station A by a cable of length 842 
ft. inclined 05 to the horizontal. 
In a vertical plane with the balloon 
and its station and on the opposite 
side of the balloon from A a target B 
was sighted from the balloon on a 
level with A. If the angle of depres- 
sion of the target from the balloon is 4, find the distance from the 
target to a point C directly under the balloon. 

18. A straight line AB on the side of a hill is inclined at 15 to the 
horizontal. The axis of a tunnel 486 ft. long is inclined 2825' below 
the horizontal and lies in a vertical plane with AB. How long is a 
vertical hole from the bottom of the tunnel to the surface of the hill? 

19. A lighthouse standing on the top of the cliff shown in Fig. 33 is 
observed from two boats A and B in a vertical plane through the light- 
house. The angle of elevation of the top of the lighthouse viewed from 
B is 16, and the angles of elevation of the top and bottom viewed 
from A are 40 and 23, respectively. If the boats are 1320 ft. apan, 
find the height of the lighthouse and the height of the cliff. 




> 32t 



92 



THE RIGHT TRIANGLE 



[CHAP. IV 




1320' 



FIG. 33. 



20. The church A and the lighthouse B represented in Fig. 34 were 
observed from a ship at point S to be on a straight line passing through S 




FIG. 34. 

and bearing N. 15 W. After sailing 5 miles on a course N. 42 E., the 
captain of the ship found that A bore due west and B bore N. 40 W. 
Find the distance from the church to the lighthouse. 

fej) A tower (Fig. 35) of height h stands on level ground and is due 
north of point A and due east of point B. At A and B the angles of 




elevation of the top of the tower are a and ft respectively. If the 
distance AB is c, show that 



Vcot 2 a + cot 2 ft 



39] 



MISCELLANEOUS EXERCISES 



93 



22. Given the oblique triangle ABC 
of Fig. 36 in which A, B, and a are 

known. Show that b = -j -r sin B. 
sin /i 

Hint. Drop a perpendicular p from 
the vertex C to the side AB. Find 
two values of p and equate them. 

23. In the oblique triangle ABC 

(Fig. 37) show that cos A = 
52 _j_ C 2 _ a 2 




AD = 6 cos A, and 

DB = c b cos A. Equate two Fio 37 

values of p. 

24. If R is the radius of a circle, show that the area of a regular 

180 
circumscribed polygon of n sides is A = nR* tan 

(25) Show that the area of a regular polygon of n sides each of length a 

, , na* 180 
is given by A = -j- cot ~~:r~* 



CHAPTER V 
FORMULAS AND GRAPHS 

40. Introduction. In Chap. Ill definitions of the trigono- 
metric functions applicable to an angle of any magnitude were 
given. In this chapter formulas based on these definitions arc 
deduced, and the graphs of the trigonometric functions are dis- 
cussed and drawn. A new unit of angular measure, the radian, is 
introduced at this point. It will be used in connection with the 
graphs and in various places throughout the text. 

41. The radian. There is a unit of angular measurement used 
so frequently in higher mathematics that it is understood to be 
the unit of measurement when 110 other is specified. Its impor- 
tance is due to the fact that various mathematical expressions 
take simpler forms in terms of this unit than in terms of any other. 
For this reason we consider it in trigonometry. This unit is 
called the radian. 

The angle subtended at the center of a circle by an arc of the circle 
equal in length to its radius is called a radian. A chord of a circle 
equal in length to its radius subtends an angle of 60 at its center; 
an arc on the same circle equal in length to its radius would sub- 
tend at its center an angle slightly less. Therefore an angle of 1 
radian is slightly less than 60. In fact, since the circumference 
of a circle is 2irR, the length of the radius is contained in the 
length of the circumference 2ir times. Hence, since the complete 
circumference subtends 360, 2w radians (= 6.2832 radians) are 
equivalent to 360. Accordingly we write 

27r radians = 360, or w radians = 180. (1) 

Since ir radians are equivalent to 180, 1 radian is l/V times as 
much; that is, 

1 radian = ( ) = 57.2958 = 5717'45". (2) 

\ 7T / 

94 



41] THE RADIAN 95 

Also, from (1), 180 is equivalent to TT radians; hence 1 is equiva- 
lent to 1/180 times TT radians. Accordingly, we write 

1 = ~ radian = 0.017453 radian. (3) 

loU 

From formulas (2) and (3) it appears that to find the number of 
degrees in a given number a of radians multiply a by 180/Tr, 
and to find the number of radians in a given number b of degrees 
multiply ft by 7T/180. 

By way of illustration, we write 

10 = 10 KT^:) radian = -^ radian; 
yioU/ lo 

5 7T . 7T 

= 60 180 radlau = 2160 



0.75 radian - 0.75 I - - 1 - 42.9719 = 4258 / 19 // . 

EXERCISES 

1. Express the following angles in radians: 

() 45. (d) 180. (g) 2230'. 

(b} 60. (r) 120. (h) 200. 

(c) 90. (/) 135. (i) 480. 

2. Express the following angles in degrees: 

(a) 7T/3 radians. (c) ir/72 radian. (e) 20rr/3 radians. 

(b) 37T/4 radians. (d) "TT/O" radians. (/) 0.987T radians. 

3. Express in radians the following angles accurate to four significant 
figures : 

(a) 1. (c) 1". (e) 18034'20". 

(b) l f . (d) 1011'25". (/) 30025'43". 

4. Find, accurate to the nearest minute, the following angles in degrees 
and minutes: (a) ^o radian; (b) 2j radians; (c) 1.6 radians; (d) 6 radians. 

5. Evaluate the following (without tables) : 

(a) tan JTT. (d) tan ^TT. (g) cot ^TT. 

(b) sin -ST. (r) sin ^TT. (h) sec ^TT. 

(c) cos r". ( /) COSTT. (i) tan (-TT). 



96 FORMULAS AND GRAPHS [CHAP. V 

6. Find the number of radians through which each of the hands of a 
clock turns in (a) 5 min., (b) 15 min., (c) 45 min., (d) 2 hr., 
(e) 6 hr. 30 min. 

7. Find the values of x and y in x = 2(6 sin 6) andy = 2(1 cos 6) 
when (a) = 0, (b) 6 = fa (c) 6 = fa (d) 6 = f TT, (e) = fa (/) 
= fa (9) = JT, (h) = TT, (i) 6 = |TT, tf) * = 27r, (/b) (9 = 7ir. 

8. If x = 5(cos + sin 0) and y = 5(sin - cos 0), find the 
value of x and y when (a) = 0, (6) = -j^r, (c) = ^TT. 

9. Two angles of a triangle are ^TT and . Find the third angle in 
sexagesimal units. 

42. Length of circular arc. Figure 1 shows a central angle of 
1 radian and a central angle of radians in 
a circle of radius r. Since two central 
angles in a circle have the same ratio 
as their intercepted arcs, we have 

6 = s 
1 ~ r 
or 
FIG. i. s = r6 units. (4) 

Example 1. A target in the form of a circular arc having its 
center at a gun is 3000 yd. from the gun and subtends at the gun 
an angle of 0.015 radian. Find the length of the target. 

Solution. Here r = 3000 yd., and = 0.015 radian. Substi- 
tuting these numbers in (4), we obtain 

8 = r6 = 3000(0.015) = 45 yd. 

Example 2. The nautical mile, or sea mile, used in the United 
States is the arc length subtended on a circle of diameter 
7917.59 miles by a central angle of 1' (7917 miles is approximately 
the diameter of a sphere having a volume equal to that of the 
earth). Find the length of the nautical mile accurate to five 
figures. 

Solution. Using formula (4) with 

r = 4(7917.6) (5280) and = -^ X ^ 
we obtain 

S = i(7917.6) (5280) -- - 6080.4 ft. 




42] LENGTH OF CIRCULAR ARC 97 

* 

This is approximately the length of the nautical mile. A more 
accurate value is 6080.27 ft. 

EXERCISES 

1. For a circle of radius 720 ft., find the length of arc subtended by a 
central angle of (a) 18; (b) 2830'; (c) 1720'30"; (d) 20'30"; (e) 38"; 
(/) (a/r). 

2. For a circle having a circumference 3000 ft. in length, find in 
degrees, minutes, and seconds the central angle subtended by an arc of 
length (a) 300 ft.; (b) 10 ft.; (c) 1 ft.; (d) 12 ft.; (e) 2807 ft. 

3. Show that a central angle of 6 degrees subtends on the circum- 
ference of a circle of radius r a length s given by 



_ - 
180 ~ TTT' 

4. If a circular arc of 30 ft. subtends 4 radians at the center of its 
circle, find the radius of the circle. 

6. If two angles of a plane triangle are respectively equal to 1 radian 
and ^ radian, express the third angle in degrees. 

6. An enemy battery 6000 yd. distant from an observation post 
subtends at the post an angle of sV radian. How many yards of front 
does the battery occupy if the post is directly in front of it? 

7. Find approximately the angle in radians subtended by a church 
spire 160 ft. high at a point in the horizontal plane through the base of 
the spire and distant 1 mile from it. 

8. An automobile whose wheels are 34 in. in diameter travels at 
the rate of 25 miles per hour. How many revolutions per minute does 
a wheel make? What is its angular velocity in radians per second? 

9. A mil* is yeVo of a right angle. Find the fraction of a radian in 
1 mil and the number of mils in 1 radian. 

10. A mil is approximately the angle subtended at the center of a 
circle having a radius of 1000 yd. by an arc length of 1 yd. on the circle. 
If for a circle r and s are expressed in yards and in mils, prove that 



8 = 



11. An enemy battery, range 6000 yd., subtends an angle of 12 mils. 
How many yards of front does it occupy (see Exercise 10) ? 

12. A grade is the hundredth part of a right angle. Express an angle 
of 1 grade in radians. Also show that a mil is iV of a grade. 

* For a discussion of the mil, see Appendix A. 



98 FORMULAS AND GRAPHS [CHAP. V 

13. Assuming the earth to be a perfect sphere 7917 miles in diameter, 
find the length of an arc on the equator that subtends an angle of 1 
at the center of the earth. Also find the distance between two points 
on the same meridian if one is 8 north of the equator and the other 
530' south of the equator. 

14. When the moon is 289,000 miles from the earth, its diameter sub- 
tends about 31' of angle at a point on the earth. Using this fact, com- 
pute the diameter of the moon by assuming that the diameter is the 
arc of a circle having its center at a point on the earth. 

16. The larger of two wheels about which a belt is drawn taut has a 
3-ft. radius. If the centers of the wheels are ft. apart, and if the 
arc of the larger wheel in contact with the belt subtends at its center an 
angle of 3.4 radians, find the radius of the smaller wheel. 

16. An automobile has tires 28 in. in diameter. Find the angular 
velocity in radians per second of the wheel of the automobile when 
going 50 miles per hour. 

17. The drive wheel of a locomotive is ft. in diameter. Find its 
angular velocity in radians per minute when the train is moving 00 
miles per hour. 

18. The drive wheel of a locomotive is ft. in diameter. If it makes 
f)00 radians per minute, find the speed of the train in mile$ per hour. 

19. Find the average speed of a man who runs two laps in 30 sec. 
on a circular track that is 35 ft. in diameter. 

In exercises 20 to 25, give approximate answers based on formula (4). 

20. On approaching the shore, the captain of the ship shown in 
Fig. 2 measured the angle of elevation of the top of a flagstaff and 




FK;. 2. 

found it to be 2 10'. If he knew the height of the staff was 32 ft. and 
if the foot of the staff was on the same level with the captain's eye, find 
his distance from the flagstaff. 

21. A lighthouse 100 ft. high stands on a rock. From the bottom of 
the lighthouse the angle of depression of a ship is 247', and from the 



43] FUNCTIONS OF 90 - $ 99 

top of the lighthouse its angle of depression is 42'. What is the height 
of the rock? What is the horizontal distance from the lighthouse to 
the ship? 



22. The signal-corps man shown ^^,^^ 90* 
in Fig. 3 subtends an angle of 35' 85^-*^ __^ - 

at station *S Y . If he is 6 ft. tall, ^^^-^-^"^ 
find his distance from the station. S 

FIG. 3. 

23. In approaching a fort situated on a plain, a reconnoitering party 
finds at one place that the fort subtends an angle of 3 and at a place 




FIG. 4. 

200 ft. nearer the fort that it subtends an angle of 6. How high is the 
fort, and what is the distance to it from the second place of observa- 
tion (see Fig. 4)? * 

24. The line of sight of a gun passes through a target 10,000 yd. away. 
Through an error in the sighting mechanism of the gun the plane of fire 
makes an angle of 10 mils with the vertical plane through the line of 
sight. How far from the target will the shell burst occur if the gun is 
correctly elevated? 

25. Statistics show that when a shell bursts within 50 ft. of an air- 
plane it registers an effective hit. Find, for effective shooting, the 
maximum deviation from the direction that would give a central hit on 
an airplane distant 10,000 yd. Assume the airplane extends through 
a circle of diameter 75 ft. 

43. Functions of 90 0. The trigonometric functions of 
90 have been expressed in terms of when 8 is acute. We 
shall now show that these same expressions hold true when 
is any angle. 

In Fig. 5, OX and OY represent rectangular coordinate axes 
and angle C\OB\ represents an acute angle 0. From Bi on the 
terminal side of angle XOB\, BiCi is drawn perpendicular to the 
.r-axis. Angle XOB is drawn equal to angle 90 0, OB is taken 
equal to OB\, and BC is drawn perpendicular to the z-axis. In 
Fig. 6 angle 6 represents an obtuse angle; in Fig. 7, angle is 



100 



FORMULAS AND GRAPHS 



[CHAP. V 



greater than 180 but less than 270; and, in Fig. 8, angle is 
greater than 270 but less than 360. The description of Fig. 5 
given above applies also to Figs. 6, 7, and 8 except in the state- 
ments of the magnitude of the angle 0, 
The two triangles OCiBi and OCB in 
each of Figs. 5, 6, 7, and 8 arc equal 
since in each case they have the 
hypotenuse and an acute angle of one 
equal, respectively, to the hypotenuse 
and an acute angle of the other; hence, 
-X in each figure, OB = OB 1} OC = CiB lt 
CB = OCi. 

Now let us agree that a line segment 
MN parallel to the y-axis is positive when a point moving on 
this line from M to N is moving in the positive direction of the y-axis, 

Y 




Fm. 5. 





FIG. 6. 



FIG. 7. 



FIG. 8. 



and negative when a point moving from M to N is moving in the 
negative direction of the y-axis. Thus in Fig. 5 the positive 
direction of the ?/-axis is toward the top of the page; hence 

segments CiBi and CB are positive, 
but the same segments when read 
BiCi and BC are considered nega- 
tive. Let us agree that a line segment 
MN parallel to the x-axis is positive 
when a point moving on this line from 
M to N is moving in the positive direc- 
X tion of the x-axis, and negative when a 
point moving from M to N is moving 
in the negative direction of the x-axis. 
Thus in Fig. 5 the positive direction of the z-axis is to the right; 
hence segments OCi and CCi are positive but the same segments 
when read C\0 and C\C are considered negative. Referring to 




44] 



FUNCTIONS OF 90 + 0, 270 + 6, 180 0, -0 



101 



Fig. 5, we should write dO = -OCi, CiC = -CCi, EC = -C, 
and Ci#i = BiCi. A line segment forming a hypotenuse will be 
considered positive in all cases. 

From Fig. 5 we read in accordance with the definitions of the 
trigonometric functions: 



w 



C* /? OC* 
sin (90 - 9) = yfi = -^i =* cos 0, 

/^/nr /nr ?j 

cos (90 - 0) = = -^ * = sin 0, 



tan (90 - 0) = 



CB OC 
OC C\B! 
OC 



cot (90 - 0) = gj = ^! = tan 0, 

sec (90 - 0) = ^ = % = esc 0, 
C/C C i/5i 



05 Ofii 



csc (90 - 0) = = 



C# ~ OC, 



= sec 



(5) 



If, while reading any equation of the group (5), wo consider the 
line segments involved as applying to Fig. 6, Fig. 7, or Fig. 8, 
we find that the argument holds good in each case. Moreover, 
the argument will still hold good in the case of each figure* if 
angle 6 represents the indicated angle increased or decreased 
by any number of revolutions; this is true because changing 
the angle 6 by any number of revolutions will not change the 
line segments of the figure in any way. Hence equations (5) 
are true for all values of 0. 

44. Functions of 90 + 0, 
270 + 6, 180 8, -8. In 

the remaining cases we shall 
make the argument only for 
an acute angle. However, the 
directions for drawing the fig- 
ures and the statements made 
will apply for all angles 6. For 
each case considered bellow, the student may construct figures for 
angle 6 in elifferent quadrants, use the same letters for corre- 
sponding positions as are used in the given figure, and note that the 
statements made apply to his figures as well as to the given one. 




FIG. 9. 



102 



FORMULAS AND GRAPHS 



[CHAP. V 



In Fig. 9, OX and OY represent rectangular axes of coordinates, 
angle XOBi represents angle 0, and angle XOB represents 
90 + 6. BI is any point on the terminal side of angle 0, and 
B is taken on the terminal side of 90 + so that OB = OB\. 
The lines BiCi and BC are drawn perpendicular to the z-axis 
and meet it in points C\ and C, respectively. Since the triangles 
OBiCi and OBC are equal, OCi = CB and CO = Cifii. Hence 
from Fig. 9, we obtain 

sin (90 + 9) = = Ql = cos 0, 



cos (00 + 0) = g = -ffip- = - sn e, 
tan (90 + 0) = ^ = :r^- = - cot 9, 
cot (90 + 9) = TTp = ~ ' ' = - tan 6, 



sec (90 + 0) = 



= - esc 



esc (90 + 0) = = = sec 



(6) 




Since the construction of the 
figures for the remaining cases 
is similar to the constructions 
already explained, their de- 
scription will be omitted. 



From Fig. 10 we obtain 

sin (270 + *) = gf = -= 



cos 0, 



cos (270 



OB ~ OB, ~ sm *' 



tan (270 + 0) = ^ = -^- J - - cot 0, 



(7) 




44] FUNCTIONS OF 90 + 0, 270 + 0, 180 0, -0 

y 

and the other three formulas B 
may be obtained from these 
by using the reciprocal rela- 
tions (1) of 11. 

From Fig. 11 we obtain 

sin (180 - 6) = ^ = T^- 1 = sin 6, 

Uif Ut>\ 

nc OC 

COS (180 - 6) - m = -gg-1 = - COS 0, 

tan (180 - 0) = -fjg = _^ = - tan 0, 

Y 



and the other three formulas 
may be obtained from these by 
using the reciprocal relations. 



From Fig. 12 we obtain 

sin (180 + 0) = j| 
cos (180 + ) = g 
tan (180 + 6) = SS 



103 



FIG. 12. 



and the other three formulas may 

be obtained from these by using 
the reciprocal relations. 



(8) 





(9) 



Fio. 13. 



104 FORMULAS AND GRAPHS [CHAP. V 

From Fig. 13 we obtain 

or or* 

/ \ \J\j _ 1 a flft^ 

V, ) - jjft ~ Qjj^ - COS , { ) 

CB C\R\ 



and the other three formulas may be obtained from these by 
using the reciprocal relations. 

45. Functions of (fc 90 8). Observing the formulas (5), 
(6), and (7) and afterwards the formulas (8), (9), and (10), we 
perceive the truth of the following statements : (a) each of the six 
trigonometric functions of k 90 0, k odd, is numerically equal 
to the co-function of 0; (b) each function of k 90 6, k even, is 
numerically equal to the same function of 0; (c) the sign to be placed 
before the resulting function of 6 is the same as the sign of the origi- 
nal function in the quadrant of k 90 0, where is thought of as an 
acute angle. 

While these rules are convenient, the student will find that he 
can draw a rough figure and easily deduce from it the required 
results. 

EXERCISES 

1. Draw the four figures relating to the formulas connected with 
90 + 0', Fig. 9 is the first figure, in the second one should represent 
an obtuse angle, in the third one should represent an angle greater 
than 180 but less than 270, and in the fourth one should represent 
an angle greater than 270 but less than 360. Letter your figures to 
correspond with Fig. 9 and note that the statements made in group (6) 
apply to each of your figures. 

2. Prove formulas like those in group (0) for 270 + 0- 

3. If the angles of a triangle are A, B, and (7, express each trig- 
onometric function of A + B in terms of a function of C. Do your 
formulas hold true in each of the cases: 

< A + B < 90, A + B = 90, 90 < A + B < 180? 

4. Derive formulas expressing vers (180 + 0), vers (270 0), 
hav (360 - $), hav (-0), covers (90 + 0), covers (180 - 0) in 
terms of trigonometric functions of 0.* 

* For definitions of vers 0, hav 0, and covers 0, see (8), 4. 



45] FUNCTIONS OF (k 90 0) 105 

6. Express as functions of a positive angle less than 90: 

(a) cos 170. (d) cos (-20). 

(b) tan 110. (e) tan (-80). 

(c) cot 100. (/) sin (-120). 

6. Express as functions of 0: 

(a) sin (810 - 0). (e) tan (0 - 180). 

(b) tan (3()() - 0). (/) sec (-180 - 0). 

(c) cot (270 + 0). (g) esc (-630 + 0). 

(d) sin (0 - 90). (h) cos (990 - 0). 

7. From the table of natural functions on page 69 find sine, cosine, 
tangent, and cotangent of 

(a) 10015'. (c) 109710'. (e) 75053'. 

(b) -3953f>'. (d) -37010'. (/) -10018'. 

8. Simplify 

, . cos (90 + A) , sin (90 + A) cot (90 + A) 

sin ( -1) cos ( A) tan ( A) 

(b) cos (270 - 0) sin (180 - 0) - cos (180 + 0) sin (270 + 0). 
, , cos 2 (180 + 0) _ cos (270 - 0) 
{C) " sin 2 (-0) sin (180~0)' 

, /A cos (180 + 0) + sin 3 (-0) 



v ' sin (270 - 0) cos (270 + 0) 

,v cot (270 -|- 0) tan (180 _- 0) esc (360^~_0) 

^ e) cot (270 - 0) tan (180 + 0) sec (360 ~+T) 

9. Find the value of 

(a) sin 480 sin ()90 + cos (-420) cos (>00. 

(b) tan '.^ tan w + cot ( ~ ) cot ( - 



10. Prove each of the following: 

(a) cos 230 cos 310 - sin (-50) sin (-130) = -1. 
(6) tan 110 cot 340 - sin 160 sec 250 = esc 2 20. 

11. Find the numerical value of 

tan I!* - 2 sin - | CBC J - 4 cos^- 



106 FORMULAS AND GRAPHS 

12. Find the numerical value of 

237T 



[CHAP. V 



vers 



- covers + hav t 
36 



13. Simplify 



cos (sir + x) sin (TT x) tan (far x) 

cos (fr + x) cos (^TT + x) tan (TT x}. 

14. Find the value of each of the following expressions: 



(a) tan 3 600. (c) sin 2 

(b) <;os 3 1020. (d) cot 3 



(e) tan [(2n + DTT - *]. 
(/) cos[(2rc- DTT + JTT]. 



16. Prove 

(a) cos (TT a*) 4~ tan (TT + x) sin ( 



sec (TT -|- x). 



(b) sin (^ - e) - - - sin (2? 4 0} 

(c) cos ^ cos 6 + sin 2^ sin - cos (^ - 

(d) cos (' + x) cos (TT - J-) + sin ^ | ^) sin (T + x) - 0. 



(/) sin (90 + 0) sec (270 - 0) = tan (270 + 0). 

cos (270 + 0) _ 1 - cos (-0) 
1 - cos (180 - 0) cos (90 - 0)' 



(0) 



E G 




FIG. 14. 



16. Express the lengths of the 
line segments AD, OA, BC, FG, 
OC, and OG in Fig. 14 in terms of 
if radius OD is 1 unit. Draw 
figures analogous to Fig. 14 showing 
as (a) a second-quadrant angle; 
(6) a third-quadrant angle; (c) a 
fourth-quadrant angle. Do the line 
values of Fig. 14 apply in the 
analogous figures? 



46. Graph of y = sin x. Tho graphs of the trigonometric 
functions are important in that they picture the variations of 



46] 



GRAPH OF y ^ sin x 



107 



these functions and, at the same time, show plainly their periodic 
nature. 

First consider the graph of y = sin x. Using the table of 
values of trigonometric functions in 29 and using the formulas 
for expressing the trigonometric; functions of any angle in terms 
of functions of an acute angle, we make Table A : 

TABLE A 



x 


x rad. 


y = sin x 











30 


7T/6 


0.5 


60 


7T/3 


0.866 


90 


T/2 


1 


120 


27T/3 


866 


150 


57T/6 


5 


180 


7T 






x 
210 


x rad. 

7ir/6 


y sin x 
-0.5 


240 


47T/3 


-0.866 


270 


37T/2 


-1 


300 


57T/3 


-0.866 


330 


llTT/6 


-0.5 


360 


27T 












In Fig. 15 are represented the rectangular axes OX and OY. 
The plotting unit on the x-axis represents ir/6 radian of angle, 
and three intervals represent the unit of measure to be used in 
laying off values of y = sin x along lines parallel to the 2/-axis.* 
Plotting points on these axes to correspond with the pairs of 
values exhibited in Table A and connecting these points with a 
smooth curve, we obtain the graph shown in Fig. 15. By 
extending Table A indefinitely for values of x greater than 2w 
and for negative values of x and by plotting the corresponding 
points and drawing the curve through them, we should obtain 
both on the left and on the right of the graph drawn in Fig. 15 
curve after curve, each having exactly the same form as the 
portion shown. 

We know that sin (2ir + x) = sin x\ hence we conclude that 
when x, starting from any value, varies through 2ir radians, sin x 



* The unit of measure used for abscissas is not necessarily the same as the 
unit for ordinates. 



108 



FORMULAS AND GRAPHS 



[CHAP. V 



varies and takes on all of its possible values once. We express 
this fact by saying that sin x is periodic and has the period 2ir. 



/ 




































































































































^ 


<r 


F 


r * 


> 
































^ 




i 






i 

i 


s 


V 


























y 


i 
i 




i 






i 




\ 


y 




















/ 


A 


? 







Is 


2 


r 


2 


5 


w_ 


7T 


V 








1 








^^ , 


ITT 






6 




3 


"2 




3 


-( 


> 




^ 


< 






1- 


i 




X 
































ks 




1 


x 


X 





























































































Fiu. 15. 




FIG. 16. 

Figure 16 shows the part of the curve y = sin a* corresponding 
to a change of three periods in x. 

47. Graph of y = cos x. Using the table of values of trigono- 
metric functions in 29, and using the formulas for expressing 
the trigonometric functions of any angle in terms of functions 
of an acute angle, we make Table B. 

Plotting the points to correspond with the pairs of values 
exhibited in Table B and connecting these points with a smooth 
curve, we obtain the graph shown in Fig. 17. The complete 
graph of y = cos x consists of an endless undulating curve extend- 
ing both to the right arid to the left of the graph drawn in Fig. 17.* 

Since cos (2w + x) = cos x, we conclude that cos x is periodic 
and has the period 2w. 

* Since cos x = sin f5 x ), it appears that the cosine curve has tho 

same form as the sine curve. In fact, if the cosine curve be translated as a 
whole 7T/2 units parallel to the s-axis, it will coincide with the sine curve. 



48] 



GRAPH OF y = tan x 



109 



o 



y 



-1 



FIG. 17. 
TABLE B 



x 


x rad. 


I/ = COS X 








1 


30 


7T/6 


0.866 


60 


7T/3 


0.5 


90 


IT /2 





120 


27T/3 


-0.5 


150 


57T/6 


-0.866 


180 


IT 


-1 



x 


x rad. 


?/ = cos a; 


210 


77T/6 


-0.86(5 


240 


47T/3 


-0.5 


270 


37T/2 





300 


57T/3 


0.5 


330 


llTT/6 


0.866 


360 


27T 


1 









48. Graph of y = tan x. The Table C of values applies to 
y = tan x, and Fig. 18 shows the corresponding graph. The 
straight lino perpendicular to the a>axis at x = ir/2 is drawn to 
indicate that, as the abscissa of a moving point on the curve 
approaches ir/2 as a limit, the point on the curve approaches 
indefinitely close to the line, and the length of the ordinate of 
the point becomes greater and greater without limit. The other 
line perpendicular to the x-axis where x = 3?r/2 indicates the 
same kind of situation. Both the table of values and the graph 
show that the part of the curve from TT to 2-n- has the same form 
as the part from to TT. This follows also from the fact that 



110 



FORMULAS AND GRAPHS 



[CHAP. V 



r 



FIG. 18. 



TABLE C 



x 


x rad. 


i/ = tan x 











30 


7T/6 


0.577 


60 


T/3 


1.732 


90 


T/2 


00 


120 


27T/3 


-1.732 


160 


5ir/6 


-0.577 


180 


ir 






x 


x rad. 


y = Ian x 


210 


77T/6 


0.577 


240 


47T/3 


1 732 


270 


37T/2 


00 


300 


57T/3 


-1.732 


330 


llTT/6 


-0.577 


360 


2w 












49] 



GRAPHS OF y - cot x, y = sec x, y = csc x 



111 



tan x = tan (TT + z). The complete curve consists of an endless 
number of branches having the same form as the branch corre- 
sponding to the values of x from ir/2 to 3w/2. From this 
discussion it appears that tan x is periodic and has the 
period TT. 

49. Graphs of y = cot *, y = sec x, y = esc x. The graphs 
of y = cot x (see Fig. 19), y = sec x (see Fig. 20), and ?/ = rsc x 







5 



2JT 



* y * cot x 

FIG. 19. 

(see Fig. 21) are obtained from the sets of values shown in the 
following table. 

In every case the complete graph consists of an endless number 
of parts, each congruent with the part shown. 

It is easily seen that each of the functions graphed has the 
same period as its reciprocal function. 



50] 



GRAPHS AND PERIODS 
TABLE D 



113 



x 


xr&d. 


y cot a; 


y = sec x 


y = esc x 








00 


1 


00 


30 


/6 


1.732 


1.155 


2 


60 


7T/3 


0.577 


2 


1.155 


90 


IT/2 





00 


1 


120 


27T/3 


-0.577 


-2 


1 155 


150 


57T/6 


-1.732 


-1.155 


2 


180 


7T 


00 


-1 


00 


210 


77T/6 


1.732 


-1.155 


-2 


240 


47T/3 


0.577 


-2 


-1 155 


270 


37T/2 





00 


-1 


300 


57T/3 


-0.577 


2 


-1 155 


330 


llTT/6 


-1 732 


1 155 


2 


360 


27T 


00 


1 


00 



50. Graphs and periods of the trigonometric functions of kO. 

First consider the graph of y sin 2x. The Table E of values 
is found as in the preceding articles. Plotting the corresponding 
points and connecting them with a smooth curve, we have 
Fig. 22. From Table E as well as from Fig. 22 it appears that 
?/(= sin 2x) has taken its complete set of values twice, once 
while x passed from to w and once while x passed from IT to 2ir. 
Hence we conclude that the period of sin 2x is 2ir/2 = TT. Since 
2x passed through 2ir radians while x passed through TT radians, 
the period of sin 2x is one-half the period of sin x. Similarly 
it appears that kx would pass through 2ir radians while x passed 
through 2ir/k radians; hence the period of sin kx is 2ir/k. A like 
argument would show that the period of cos kx is 2rr/k, the period 
of tan kx is ir/k, and each reciprocal function has the same period 
as the function of which it is the reciprocal. 



114 



FORMULAS AND GRAPHS 



[CHAP. V 



In plotting y = sin kx and y = cos for, we observe that the 
greatest value that y may have is unity. Evidently, if we should 
Y 



' 
















































































S 


\ 














/* 


\ 


















/ 


\ 


i 
1 


\ 










/ 






\ 


3r 
T 














f 


\ 
I 


i 


? 








1 






\ 








2rr 


^ Y 


o 




n 


7T 




\ 






1 


IT 








v 






1 
















\ 






1 










\ 






1 


















v^ 


s 














^ 


^/ 


























y=sin2jc 














FIG. 22. 
TABLE K 



x rad. 


x 


2x 


y = sin 2x 














7T/6 


30 


60 


0.866 


7T/3 


60 


120 


0.866 


7T/2 


90 


180 





27T/3 


120 


240 


-0 866 


57T/6 


150 


300 


-0.866 


T 


180 


360 





77T/6 


210 


420 


0.866 


47T/3 


240 


480 


0.866 


37T/2 


270 


540 





57T/3 


300 


600 


-0.866 


llTT/6 


330 


660 


-0.866 


2r 


360 


720 






plot y = a sin kx or y a cos kx, the greatest value y could 
attain in either case would be a. This number a is spoken of 
as the amplitude of y. 



50] GRAPHS AND PERIODS 115 

EXERCISES 

1. Find the period of each of the following functions: 

(a) sin 50. (h) 5 tan 7T0. 

(b) 3 cos 80. (i) 3 cot ^. 

(c) 2 tan 0. (j) 7.9 sec (3? - 45). 

(d) i cot 40. (k) 2 + sin 3<p. 

(e) 2 sec 60. (0 6 + cos 2<p. 
(/) 242 esc 20. (m) -6 tan <p. 

(0) 5 cos (40 + 60). (n) 112 sin (2770 + 30). 

2. Find the amplitude of each of the following functions: 

(a) sin (V- (e) 334 cos (<p + 60). 

(b) 4 cos 6<p. (/) i^ cos (<p TT). 

(c) 2 sin !<>. (0) cos (2 + 0). 

(d) cS.<> cos ^. (h) 8 sin (2410 - 45). 

3. Plot: 

2x 

(a) // cos .r. (/) ?/ f> sec .r. (k} y sin -77- 

x 

(/>) ?/ 2 sin r. (0) // 2 sin 2a\ (/) iy ~ cos v 

/ 

(r) y = 2 tan a*. (/i) // = 4 tan 2.r. (m) 2?y = cot 7- 

(rf) ?/ = 3 cot x. (i) 2y = cos 2x. (n) y = sec (re + TT). 

(f) y - 4 esc re. (j) y = tan jj:. (o) 2/ = esc f ^ + 0V 

4. Plot on the same set of axes: 

(a) y = cos re and y = cos 2rc. 

(b) y = sin re and y = 2 sin re. 

(c) y = tan re and y = cot re. 

(d) y = 2 sin re and y = 2 esc re. 

(e) y = sin 2rc and y = cos rc. 
(/) y = 2 tan 2re and i/ = cot Ire. 

5. Plot the graph of each of the following equations for the indicated 
range of values of re: 

(a) y = sin re + cos re, to 2?r. 

(6) y = 3 cos x + 2 sin re, TT to 2tr. 

(c) y = cos re + 3 sin 2rc, TT to TT. 

(d) y = sin re cos x, TT to IT. 

(e) y = sin re 2 cos re, 2?r to 2ir. 



116 FORMULAS AND GRAPHS [CHAP. V 

6. By plotting the graph of y = sin x and using esc x = I/sin x, 
obtain the graph of y = esc x on the same set of axes and to the same 
scale. 

7. By plotting the graph of y = cos x and using sec x = I/cos x, 
obtain the graph of y = sec x on the same set of axes and to the same 
scale. 

8. Plot the curve y = sin '3x. Then construct the curve y = esc 3x 
on the same graph by. taking account of the fact that esc 3x and sin 3x 
are reciprocal functions. 

9. Plot one period of the graph of each of the following equations 
on the same set of axes and to the same scale: 

(a) y sin x, y = sin 2x, and y = sin ^x. 

(b) y = sin x, y = 2 sin x, and y = sin x. 

(c) y = cos Xj y = cos 2z, and y = 2 cos x. 

(d) y = cos x, y = ^ cos x, and y = cos f x. 

10. If t stands for time in seconds and y for magnitude in volts, then 
the equation 

y = 110 sin 377* 

represents the voltage causing an alternating current of electricity. 
Find the period and the maximum magnitude of the voltage. 

51. MISCELLANEOUS EXERCISES 

1. Express the following angles in radians: 10, 30, 45, 135, 225, 
-270, -18, -2415'. 

2. Construct approximately the following angles: 2 radians, 
3^ radians, J radian, 4 radians, 9 radians. 

3. Construct the following angles: 

TT 7T 7T _5?T 57T 

2' ~3' 4' *' " 4 ' ~2~' 

4. Express the following angles in degrees: radians, TT radians, 

2 7 

o TT radians, T TT radians, 2 radians, 5 radians, 3 radians. 

u 4 

5. Express the following as functions of an acute angle less than 45 : 

, x , 8?r 177T 

(a) cot-7jj" (c) tan-jg- 



/IA /j\ 

(b) sin -jj- (d) sec ^ 

6. In a circle whose radius is 5, the length of an intercepted arc is 12. 
"Find the corresponding central angle (a) in radians; (b) in degrees. 



51] MISCELLANEOUS EXERCISES 117 

7. In a circle of radius 12 ft., find the length of the arc intercepted 
by a central angle of 16. 

8. Find the angle between the tangents to a circle at two points 
whose distance apart measured on the arc of the circle is 378 ft., the 
radius of the circle being 900 ft. 

9. Assuming the earth's orbit to be a circle of radius 92,000,000 
miles, what is the velocity of the earth in its path in miles per second? 

10. A belt travels around two pulleys whoso diameters are 3 ft. and 
10 in., respectively. The larger pulley makes 80 revolutions per minute. 
Find the angular velocity of the smaller pulley in radians per second; also 
the speed of the belt in feet per minute. 

11. Find the numerical value of: 

(a) cos 30 + cos 150 + tan 60 + tan 120. 

(b) (tan 120 - tan 135) X (tan 120 + tan 135). 

(c) sin 420 cos 390 + cos (-300) sin (-330). 

(d) cos 570 sin 510 - sin 330 cos 390. 

/ N 4- 2 ' 7 _!_ 3 I 5 * 

(e) tan ^TT sin TTTT + sec -pr esc 2 -5- 

O O rr 5 

(/) 3 tan 210 + 2 tan 120. 
(flf) 5 sec 2 135 - 6 cot 2 300. 

12. Simplify each of the following expressions: 

(a) cos f g + x ) ** n (& 1 " z) cos (2ir + x) sin (-<r x\ 

(b) sec (180 - 6) X cos X tan (180 - 0) X cot 0. 
cos (90 - .4) cosj4 __ tanJ270 + A) 

(c) sin (180 + A) + sin (90 + A) "*" tan (-A) 

(d) sec (180 + 0) esc (270 + 0) + tan (180 - 9) 

cot (270 - 6). 

cos (180 - 0) cot (270 + B) cos (270 - 6) 
(e > sin (90 - 0) + "~ ~ sec (-0) 

cos (90M-a) tan (-a) 
(1) ~ shi (-) " + tan (180 + a)" 

sin (180 - 0) tan (180 + 9) 
(g > cos (90 + 9) cot (90 + 0) ' 

13. Prove: 

(a) cos (90 +Wtan (180 + 0) = 1/csc (270 - 0). 
tan (180 + a) - tan (180 - 0) 

-~^ = tan a tan * 



tan 37T - tan 20 
r+ton8rtan2g 



118 FORMULAS AND GRAPHS [CHAP. V 

(d) (a - b) tan (90 - x) + (a + b) cot (90 + x) 

= (a 6) cot x (a + 6) tan x. 

(e) sin ( ? + x j sin (IT + x) + cos ( ~ + x J cos (TT x) = 0. 

(/) cos (TT + x) cos ( ~ IM sin (TT + x) sin f -y - y j = 

cos a; sin y sin # cos y. 

(g) tan x + tan ( y) tan (TT y) = tan . 

14. If cot 260 = +a, prove that cos 350 = +- ,=L 

Vl +a 2 

16. If sec 340 = +a, prove that sin 110 = -, and tan 110 = 
1 



16. If cos 300 = +a, prove that cot 120 = 7=^ = ' 

VI ~ 2 

17. Show that cot (270 + x) is equal to the negative of the cotangent 
of the supplementary angle. 

,o A oino * , sin 320 - cos 310 . . . 

18. If tan 310 = c, find lW -^.-^p m terms of c- 

19. If sin B = jy and 6 is in the third quadrant, find the functions 
of (-0). 

20. If cot ( 6) = 2 and 6 is in the second quadrant, find the functions 
of 0. 

21. If cos a = TS and a is in the second quadrant, evaluate: 

sin J180 -a) cos (360 - a) 
sec (270 + a) + esc (270 - a)* 

22. Tan /3 = f and is in the third quadrant, evaluate: 

sin (-0) esc 2 (180 + 0) _ cot (270 + 0) 
sec 2 (90 + j8) tan (180 - 0)' 

23. Plot y = sin 2x. 

24. Plot y = 3 cos z. 

25. Plot y = tan ?x. 

26. Plot t/ = cos 2x and y = sec 2x on the same set of axes. 

27. Express in radians the sum of the angles of a convex polygon of 
n sides. 

28. The rotor of a steam turbine is 2 ft. in diameter and makes 2500 
revolutions per minute. The blades of the turbine, situated on the 
circumference of the rotor, have one-half the velocity of the steam that 
drives them. What is the velocity of the steam in feet per second? 



51] MISCELLANEOUS EXERCISES 119 

29. The diameter of the sun is approximately 864,000 miles and at a 
certain instant it subtends an angle of 32' at a point on the earth. 
Compute the approximate distance from the earth to the sun at this 
instant. 

30. Assuming that the diameter of the smallest sphere clearly visible 
to the ordinary eye subtends an angle of 1' at the eye, find the greatest 
distance at which a baseball 2.9 in. in diameter can be clearly seen. 

31. A horse is tethered to a stake at the corner of a field where the 
boundaries intersect at an angle of 75. How long must the rope be 
so that the horse can graze over half an acre? 

32. Find the length in feet of an arc of 3" on the earth's equator. 



CHAPTER VI 
GENERAL FORMULAS 

62. The addition formulas. In many respects, the two 
formulas 

sin (A + B) = sin A cos B + cos A sin , 
cos (A + B) = cos A cos B sin A sin B, 

are the most important ones in trigonometry. They are called 
the addition formulas because they express trigonometric func- 
tions of the sum of two angles in terms of the trigonometric 
functions of the angles. These formulas, holding true as they do 
for all angles, positive and negative, are the basis of trigonometric 
analysis. It will appear in what follows that all the formulas of 
this chapter and many others are derived from them. 

53. Proof of the addition formulas. Special case. We shall 

first prove formulas (1) for the case 
when both angles A and B are posi- 
tive acute angles and A + B < 90. 
In Fig. 1 angles A and B appear as 
adjacent angles' with common vertex 
O and common side OC. Point D is 
taken on the terminal side of angle 
B so that OD is 1 unit long, DC is 
drawn perpendicular to OC, DG and 
CE perpendicular to OX, and FC 
perpendicular to GD. 

The proof of formulas (1) will 
consist in finding the lengths of the line segments in Fig. 1, 
writing them on the figure to obtain Fig. 2, and then reading 
the formulas from Fig. 2. The student may do this for himself 
without reading the following development. 




From Fig. 1 we read 



CD . OC D 

-r- = sin B, = C08 B. 

120 



(2) 



53] 



PROOF OF THE ADDITION FORMULAS 



121 



Angle FDC is equal to angle A because its sides are respectively perpen- 
dicular to the sides of angle A. Hence, from triangle FCD, 



FC 
CD 



= sm 



FD 
CD 



= cos 



(3) 



Replacing CD in (3) by its value sin B from (2) and multiplying both 
members of each equation by sin B, we obtain 



FC = sin A sin B, FD = cos A sin B. 
From triangle OEC, 



EC_ 
OC 



_ 
~ sm 



OE . 

OC = cos A 



(4) 



(5) 



Replacing OC in (5) by its value cos B from (2) and multiplying both 

D 




cos A cos B 
Fio. 2. 

members of each equation by cos B, we get 

EC = sin A cos B. OE cos A cos 



(6) 



Figure 2 is the result of writing on each line in Fig. 1 its value obtained 
from one of the equations (2), (4), (5), and (6). 
Noting that 

sin (A + B) = ~ = EC + FD 

and 

OG * 

cos (A + B) - = OE - FC > 



122 GENERAL FORMULAE [CHAP. VI 

we read from Fig. 2 

sin (-4 + B) = sin A cos B + cos A sin B. (7) 

cos (A + B) = cos A cos B sin A sin B. (8) 

That the formulas (7) and (8) are true for all values of A and B 
will be proved in the next article. We shall now assume that 
they are generally true and use them to obtain two other closely 
related formulas. Replacing B by B in (7) and (8), we get 

sin [A + ( B)] = sin A cos ( B) + cos A sin ( B),\ , . 
cos [A + (-)] = cos A cos (-5) - sin 4 sin (-B).] W 

In accordance with 44, 

cos (#) = cos# and sin ( /?)= sin 5. 

Replacing cos ( B) by cos B and sin ( B) by sin Z? in (9), 
we obtain 

sin (-4 B) = sin A cos B cos A sin 5, (10) 

cos (A B) = cos A cos + sin A sin B. (11) 



Example. Use (8) to find cos 75. 
Solution. Substituting 45 for A and 30 for B in 
obtain 

00375 V2.V3 A/2.1 V6 - V2 


(8), we 


00876 - 2 2 22~ 4 



EXERCISES 

1. Use (1) to find sin (A + B) and cos (A + B) if sin A = | and 
cos B = f , and if A and 5 are both acute angles. 

2. Substitute A = 30, B = 60 in (1) to obtain sin 90 and cos 90. 

3. Substitute A = 30, B = 45 in (1) to obtain sin 75 and cos 75. 
Then write the values of the trigonometric functions of 75. 

4. By using (1) find sin 105 and then find the values of the other 
trigonometric functions of 105 from a right triangle. 

6. Given that a and ft terminate in the second and in the fourth 
quadrant, respectively, and that sin a = cos ft = f , find cos (a + ft). 

6. Using the table of natural functions, find (a) sin 31 from the 
functions of 20 and 11; (6) the difference between sin (20 + 11) 
and sin 20 + sin 11. 



53] 



PROOF OF THE ADDITION FORMULAS 



123 



7. Find cos (A + B) if sin A = | and sin B = -&, A and B being 
positive acute angles. 

8. If tan x \ and tan y yj-, find sin (x + y) and cos (x + y) 
when x and ?/ are acute angles. 

9. Set B = A in (1) to obtain sin 2 A and cos 2 A in terms of sin A 
and cos A. 

10. Set A = 90 in (1) and check the result by the methods of Chap. V. 

11. Find, by using formulas (7) to (11), the sine and cosine of: 



(a) 90 + y. 

(b) 180 - y. 

(c) 180 + y. 

(d) 270 - y. 

(e) 270 + y. 

12. Show that 



(/) 360 - y. 

(g) 360 + y. 

(h) x - 90. 

(i) x - 180. 

(j) x - 270. 



(0 45 - y. 

(m) 45 + y. 

(n) 30 + y. 

(o) 60 - y. 



sin (45 



cos x sin 



13. Show that 



cos 



(210 + x) = (sin x - \/3 cos s). 



14. Show that 



r fL* t \o i \ cos a \/3 sin ct 
cos ((>() + a) = - v 

15. Find cos (210 + A) if sec A = -\/3 antl A is a second - 
quadrant angle. 

A y 

16. In Fig. 3 let OB = 1 unit and 
express all its line segments in terms of 
trigonometric functions of and (p. 
Then deduce the formulas 

sin (0 <p) = sin 6 cos <p cos sin <p y 
cos (0 ^) = cos cos ^ + sin sin ^> 




D 



Fio. 3. 



17. Show that 



sin 08 - 120) = - 



18. Show that 



sin (45 + x) 



cos x + sin x 

775 



GENERAL FORMULAS 



[CHAP. VI 



cos y sin y 
~ 



124 

19. Show that 

sin (y + 135) 

20. Show that 

cos (A B) cos (A + B) = cos 2 A sin 2 B = cos 2 B sin 2 A. 

21. Show that 

sin (x + y) cos y cos (x + y) sin y = sin z. 

22. Show that 

sin (x + 60) cos (x + 30) = sin x. 

23. Use (1) to prove that 

(a) sin 2x = 2 sin x cos z. 

(6) cos 2as = cos 2 x sin 2 x. 

(c) sin 3x = sin x cos 2x + cos # sin 2x. 

(d) sin 3# = sin 5.r cos 2z cos 5x sin 2x. 

24. Express sin 30 in terms of sin 6. 

25. Express cos 30 in terms of cos 0. 

26. Prove that 

sin (a+ft) + sin (a - 



cos (+)+ cos (a - /S) 



= tan . 



Removal of restrictions on the addition formulas. In 53 the 

angles A and B were assumed to be 
acute angles such that A + B was 
less than 90. This article is de- 
signed to show that formulas (1) 
hold true when angles A and B are 
unrestricted in magnitude and sign. 

The proof given in 53 applies 
equally well to Fig. 4. Hence for- 
mulas (1) are true when A and B 
are any two acute angles. 

Let A be an angle greater than 
90 but less than 180, and let B be 
a positive acute angle. Let 




O cos A cos B E 

FIG. 4. 



A' = A - 90. 



(12) 



Since A! and B are acute angles, formulas (1) hold true for them, and 
we have 



55] ^ ADDITION AND SUBTRACTION FORMULAS 125 

sin (A' + B) = sin A' cos B + cos A' sin B,\ 

(13) 



,} 
. J 



cos (A' + B) =s cos A' cos 5 sin A' sin B 

Replacing A' in (13) by A 90 from (12) and using the methods of 
Chap. V, we have 



sin (A' + B) = sin (A + B - 90) = - cos (A + B), 
cos (A 7 + B) = cos (A + B - 90) = sin (A + B), 

sin A' = sin (A - 90) = - cos A, 

cos A' = cos (A 90) = sin A. 



(14) 



Substituting the values of sin (A' + J3), cos (A' + B) } sin A', and cos A' 
from (14) in (13), we obtain, after slight simplification, 

cos (A + B) = cos A cos B sin A sin B, 
sin (A + B) = sin A cos /* + cos A sin #. 

Hence it appears that formulas (1) hold true when A is an obtuse angle 
and B an acute angle. 

We next let A be an angle greater than 180 but less than 270 and 
let B be an acute angle. By letting A' = A 90 and arguing as 
above, we prove that formulas (1) hold true for this new case. By 
continuing this process indefinitely we can show that (1) holds true 
when A is any positive angle and B is a positive acute angle. Again, 
letting A be any angle and B an angle greater than 90 but less than 
180, we argue as above and show that (1) holds true in this case. Con- 
tinuing this process with reference to B, we finally deduce that (1) holds 
true Y ft en A and B are any positive angles. 

If (1) holds true for any pair of positive angles A and #, evidently it 
will still hold true if A and B be decreased by any multiples of 360. 
Since any negative angle may be obtained by subtracting some multiple 
of 360 from a suitable positive angle, and since (1) holds true when 
A and B are any positive angles, it appears that (1) holds true 
when A and B represent any negative angles. Hence (1) holds true when 
A and B represent any angles. 

55. Addition and subtraction formulas for the tangent. By 

using (1), we may deduce addition formulas for the other func- 
tions. To express tan (A + B} in terms of tan A and tan B we 
have 

(A a. m sin (A + B) _ sin A cos B + cos A sin B 
tan (A+X) - cog (A + ) - cos A cos B _ gin A sin g 

(15) 



126 GENERAL FORMULAS [CHAP. VI 

Dividing numerator and denominator of the right-hand member 
of (15) by cos A cos B, we obtain 

sin A cos B cos A sin B 

A. i A i \ cos A cos # c s ^ cos B 
tan (A + ) = - -; ----- B --- = T- - nt 
cos ^ cos B sin A sin B 



__ 
cos 4 cos /? cos A cos 5 

or 

- <"> 



Since equations (1) hold true for all values of A and B, it 
follows that (16) holds true for all values of A and B for which 
tan (A + B) is defined. Replacing B by B and therefore 
tan B by tan ( B) = tan B in (16), we obtain 

/ A T>\ tan A tan B . 

tan (A - B) = 4 . . - J-T - D - (17) 

v y 1 + tan 4 tan B v ' 

Addition and subtraction formulas for the other functions could 
be obtained by a similar procedure. 

EXERCISES 

1. Express the tangent functions in (16) in terms of cotangent func- 
tions, and thus deduce that 

j. f A i n\ COt A COt B 1 

cot (A + B) = - r r- , -- r-D~' 
v ' cot A + cot B 

2. Prove the formula of Exercise 1 by starting from formulas (1). 

3. Find tan 105 in the form of radicals by using (16). 

4. Check (16) by substituting in it A = 47T/3, B = 37T/4. 

5. If tan a = f and sin ft = ^-f , find the functions of a -f ft when a is 
of the third and ft of the second quadrant. 

6. If cos a = TT an d sin ft j^-, find the functions of a ft 
when a is of the third, and ft of the fourth quadrant. 

7. If tan x = -5 and x y = 45, find tan y. 

8. If tan y = 2 and x + y = 135, find tan x. 

9. Show that 



tan(A-60')= 



1 + \/3 tan A 



55] ADDITION AND SUBTRACTION FORMULAS 

10. Show that 

tan (x + 45) + cot (x - 45) = 0. 

11. Show that 

cn' ftt _- A\ 

cot A cot B 
12.. Show that 



127 



sin A sin B' 



cot (45 - y) 
cot (45" + y) 



1 + 2 sin y cos y 
1 2 sin y cos ?/ 



13. In Fig. I let OK = 1 unit, and express all its line segments in 
terms of trigonometric functions of A and B. Then deduce formulas 
(16) and (17). 

14. Use (1), (10), and (11) to simplify 

(a) sin 3x cos 2x + cos 3x sin 2x. 

(b) cos 3x cos 2x + sin 3x sin 2x. 

(c) sin 3x cos 2x cos 3x sin 2x. 

(d) cos (x + 45) cos (45 - x) - sin (x + 45) sin (45 - x). 

(e) cos 2 x ^Ui 2 x. 

(/) sin x cos 2^- cos x sin #. 



16. Use (16) to simplify 



(a) f 



tan 3z + tan 2x 



- tan 



(6) 



2 tan a; 
- tan 2 Y 



16. Express all line segments of Fig. 5 in terms 
of 6 and v?, and from the results deduce a formula 
for sin (0 + <p) and a formula for cos (B + <p). 



17. Taking AC of Fig. 6 equal to 1 unit, 
express all line segments of the figure in 
terms of 6 and <p, and from your results de- 
duce formula (16). 

Hint. Angle BDC = <p. 

18. Taking BC of Fig. 6 equal to 1 unit, 
deduce from the figure the formula of 
P^xercise 1. 




. 6. 



128 GENERAL FORMULAS [CHAP. VI 

19. Prove the following identities: 

/ \ x fArO I A\ 1 ~t~ ^ an ^ 

(a) tan (45 + 0) = ^ ^^ 

(6) tan (45 - a;) tan (135 -) = -!. 

(c) cos (60 + x) cos (30 + x) + sin (60 + x) sin (30 + x) 



(d) cos 5z cos 3x + sin 5z sin 3x = 2 cos 2 a? 1. 
sin (a + ff) __ cot a + cut /? 

(e) cos (a -"j8) " F+ cot a cot 0' 
(/) esc 20 = cote - cot 28. 

20. The expression a sin + b cos may be written in the form 



cos 



Hence if we let tan a. b/a y we have 

a sin + 6 cos = -\/d? + fc 2 (sin ^ cos a + cos 6 sin a), 
or 

a sin + 6 cos 6 = \Xo 2 ~+~S* sin (6 + a). (A) 

Write each of the following expressions in the form (A) : 

(a) 2\/3 sin 6 + 2 cos 0. (d) 3 sin - \/3 cos 0. 

(6) a sin 9 + a cos 0. (e) 3 sin + 4 cos 0. 

(c) 7- sin + -7= cos 0. (/) \/2 cos - \/2 sin 0. 



21. Show that 

sin (A + B + C) sin A cos # cos C + cos A sin 5 cos C 

+ cos A cos B sin C sin A sin Z? sin C. 

Hint. A+B + C = (A+B)+C. 

22. Show that 

cos (A + B + C) = cos A cos B cos C sin A cos B sin C 

cos A sin B sin C Y sin A sin J3 cos C. 

66. The double-angle formulas and the half-angle formulas. 

To express the trigonometric functions of 20 in terms of functions 
of replace ^ by in the addition formulas. Thus, to find sin 20, 



56] THE DOUBLE-ANGLE FORMULAS 129 

substitute for <f> in the formula 

sin (0 + <A) = sin cos <t> + cos sin 
and obtain 

sin (0 + 0) = sin cos + cos sin 
or 

sin 28 = 2 sin 8 cos 8. (18) 

Similarly, from the formula 

cos (0 + <) = cos 6 cos sin sin # 

we obtain 

cos 28 = cos 2 8 - sin 2 8, (19) 

By using the fact that sin 2 + cos 2 = 1, we easily deduce from 

(19) 

cos 28 = 2 cos 2 8-1, (20) 

cos 28 = 1 - 2 sin 2 8. (21) 

From formula (16), we obtain 

* - 

Solving (20) for cos and (21) for sin 0, we obtain 

, /I" + cos "29 . - , /I 

cos = ,J 1 = ~> sin = J- 



2 

To get half-angle formulas, replace by ^v in (23) and obtain 

cos ^>* 
2 (24) 

+ COS <f> 

~2 ' 

The plus sign is to be used in the first formula of (24) when 
\<p is a first-quadrant* f or a second-quadrant angle, the minus 

* Since hav <f> = (1 cos ^>)/2, we have from (24) 



. <p 1 cos <p 

sm 2 - = = hav (p. 

2 2i 

f Occasionally it will be convenient to refer to an angle as belonging to a 
certain quadrant. If the initial ray of an angle extends from the origin 
along the positive X-axis, it is called a first-quadrant angle, a second-quad- 



130 GENERAL FORMULAS [CHAP. VI 

sign when \<p is a third-quadrant or a fourth-quadrant angle. 
The plus sign is to be used in the second equation of (24) when 
Jf<p is a first-quadrant or a fourth-quadrant angle, the minus sign 
when %<f> is a second-quadrant or a third-quadrant angle. 

To obtain a formula for tan -J-^, divide the first of equations 
(23) by the second to obtain 



-f- cos 
or 



COS <p /A _ N 

(25) 



The plus sign is to be used when -^ is a first-quadrant or a 
third-quadrant angle, the minus sign when \<p is a second-quad- 
rant or a fourth-quadrant angle. From (25) we also have 

(I - cosy?) (I - cosy) 1 - cos v 

= ' (26) 



Since 1 cos <p i never negative and sin <p always has the same 
sign as tan -g-<p, the right-hand member of (2(>) does not require 
the signs. 

EXERCISES 

1. If sin a = f , cos a = , find sin 2a, cos 2a, tan 2a, sin \at, 
cos 1, anil tan !. 

2. Use formulas (^24) to find sin (22) and cos (22) from the fact 
that cos 45 = 1/^/2. 

3. Verify the following identities: 

(a) cos 2x = cos 2 x sin 2 x = 2 cos 2 x 1 = 1 2 sin 2 x. 

/f . sin 2a cos 2a 

(6) 5 - ------- = sec a. 

v ' sin a cos a 

(c) cos 2 (45 + x) - sin 2 (45 + x) = - sin 2x. 

(B 0\ 2 

sin o cos ^ j = 1 sin 0. 

(e) cos 4 6 - sin 4 = cos 20. 
tan'0 



rant angle, a third-quadrant angle, or a fourth-quadrant angle according 
as its terminal side lies in the first, second, third, or fourth quadrant. 



THE DOUBLE-ANGLE FORMULAS 131 

sin 2a + sin a 



'!"+ cos a + cos 2a 
tan 28 = - 



tan "' 



(f) tan ^> = esc <f> cot ^?. 

(j) hav ^> = sin 2 \<p. 

(k) cos 6 - sin 6 = cos 20 - i sin 40 sin 20. 

4. Substitue = 2.r, ^> = a: in sin (0 + (p) sin cos ^ + cos sin ^ 
and then use the double-angle formulas to derive 

sin 3x = 3 sin x cos 2 x sin 3 x = 3 sin x 4 sin 3 x. 
6. Using a method similar to the one suggested in Exercise 4, derive . 

(a) cos 3x = 4 cos 3 x 3 cos z. 

(6) sin 4x = 4 sin z cos #(2 cos 2 # 1). 

6. Derive a formula expressing sin 4x in terms of sin x and a formula 
expressing tan 4x in terms of tan x. 

n 

1. Prpve that, if z = tan x then 

a 2z a 1 -** , . 2 

sin = -- 2 , cos = tan = . 



8. Find sin 18 in radical form. 

Hint. First write cos 3z = sin 2x where x = 18, and express both 
members in terms of sin x and cos x. Solve the resulting equation for 
sin x. 

9. If is an angle in the second quadrant and tan = ^, find 

. cot 20. cos (270 - 20). 

sin (180 - 0). esc (180 + 20). 

10. Show that 

. x 

x sln 2 , T sin x 

(a) cot 4 = - ^ W tan ** 

1 cos s 

/rNX^Ii^O /\ .L! 

(6) cot + tan = 2 esc x. (e) cot ^o; 



_ cos 



1 cos x 
2 cot x 



132 

11. (a) Show that tan 3A 



GENERAL FORMULAS 
3 tan A - tan 3 A 



[CHAP. VI 



1 - 3 tan 2 A 



,i\ au *u 4. 4. A 4 tan s(l tan 2 s) 
(6) Show that tan 4z = F-~6 tan* *+ tan's* 




12, In Fig. 7, AD bisects the angle A and 
D# is perpendicular to AB. Hence DE = 
CD. Show from the figure that 

- sin A 



FIG. 7. 

13. Find all line segments of Figs. 8, 9, and 10 in terms of 6, and write 
several identities from your figures. Verify these identities in the usual 
way. 




k- ... i 

FIG. 8. 




k- 



-i d 

FIG 9. 





14. Prove the formula for tan (a + ) 
from Fig. 11 by using line values. 



FIG. 11. 



15. Prove that in a right triangle, C being the right angle, the follow- 
ing relations are true: 



(a) sin 2A = sin 2B. 

2ab 
6 2 - a 2 ' 



(b) tan 2A 

(c) cos 2A 



c 2 



(d) cos 2A -t cos 2 = 0. 
<e) tan B = cofc A + cos C. 

,. . , Soft* - a 8 
(/) an 3A - 



57] CONVERSION FORMULAS 133 

67. Conversion formulas. From (1) and (10), we have 

sin (6 + <p) = sin cos <p + cos sin ^, 
sin (6 <p) = sin 6 cos <f> cos sin <p. 

Adding these two formulas member by member, we get 

sin (6 + <p) + sin (0 <p) = 2 sin 8 cos ^, (27) 

and subtracting the second from the first, we obtain 

sin (8 + <p) sin (0 <p) = 2 cos 8 sin <?. (28) 

From (1) and (11) we get 

cos (6 + <p) = cos 6 cos <p sin sin ^>, 
cos (0 v>) = cos 6 cos <p + sin 6 sin ^. 

Adding these formulas member by member and afterwards 
subtracting the second from the first, we obtain 

cos (8 + <p) + cos (8 - <p) = 2 cos 8 cos <p, (29) 

cos (8 + <p) cos (8 <p) = 2 sin 8 sin <p. (30) 

Formulas (27) to (30) should not be memorized but should be 
recalled by mentally carrying out their derivation from the 
addition formulas. These formulas are important because they 
enable us to express a product of sines and cosines as a sum of 
two or more expressions or to express a sum or a difference of two 
trigonometric functions in the form of a product. The following 
examples will illustrate the method of doing this. 

Example 1. Expand cos 2x cos 3x sin 4# into a sum of sines 
and cosines of multiple angles. 

Solution. Using (29) with 6 = 2x, <p = 3x, we obtain 

2 cos 2x cos 3x = cos (2x + 3x) + cos (2x 3x), 

or 

2 cos 2x cos 3x = cos 5x + cos x. (a) 

Multiplying (a) through by sin \x and dividing by 2, we get 

cos 2x cos 3x sin 4x = ^(cos bx sin 4x + cos x sin 4z). (6) 
Then using (27) with 6 = 4z, <p = 5z, we obtain 

2 sin 4x cos 5x = sin (4# + 5x) + sin (4# 5x), 
or 

2 sin 4# cos 5# = sin 9x sin #. (c) 



134 GENERAL FORMULAS [CHAP. VI 

Again using (27) with 6 = 4z, <p = x, we obtain 

2 cos x sin 4x = sin 5x + sin 3x. (d) 

Substituting sin 4# cos 5x from (c) and cos x sin 4x from (d) 
in (6), we obtain, after slight simplification, 

cos 2x cos 3x sin 4z = ^(sin 9x sin x + sin 5* + sin 3x). 



Example 2. Express sin 5x sin 3x in the form of a product. 
Solution. The left-hand member of (28) will be the desired 
difference if we set 

+ <p = 5x, 6 <p = 3.r, (a) 

or, solving for and <f> in terms of x, 

d = 4x, <p = x. (b) 

Substituting 6 and <p from (fr) in (28), we obtain 
sin 5x sin 3x = 2 cos 4x sin x. 

A process similar to that carried out in (a) and (6) to find 
and <p in terms of the given angles may be used to derive another 
set of formulas that are convenient for transforming a sum to a 
product. Let 

+ ^= a> e - <p = P. (31) 

Solving (31) simultaneously for and <p in terms of a and 0, we 
get 

= *( + ft, * = i( - 0). (32) 

Replacing by ( + 0) and ? by }( - 0) in (27), (28), (29), 
and (30), we obtain 

sin a + sin 5 = 2 sin ( + 5) cos (a - J), (33) 

sin a - sin 5 = 2 cos (a + 5) sin ( - ff), (84) 

cos a + cos g = 2 cos (a + g) cos ^(o - 5), (86) 

cos a - cos g = -2 sin (a + g) sin (a - ff). (36) 

EXERCISES 
1. Express in the form of a product 

(a) sin 35 + sin 25. (e) cos 4z + cos 2s. 

(b) sin 45 - sin 30. (/) sin 5x - sin 2x. 

(c) cos 65 + cos 25. (0) sin 3s + sin x. 

(d) cos 75 - cos 5. (h) cos 5x - cos 3z. 



57] CONVERSION FORMULAS 135 

2. Expand into a sum of sines and cosines of multiple angles : 

(a) sin 3z cos 7x. (c) sin x sin 2x cos 3z. 

(ft) cos 3x cos 7z. (d) cos 3x cos 5s sin 7x. 

Verify the following identities: 

3. sin 32 + sin 28 = cos 2. 

4. sin 50 - sin 10 = \/3 sin 20. 
6. cos 80 - cos 20 = - sin 50. 

6. cos 140 + cos 100 + cos 20 = 0. 

7. tan 50 + cot 50 = 2 sec 10. 

8. cos 60 + cos 30 = \/2 cos 15. 

9. sin 40 - cos 70 = \/3 sin 10. 

10. sin (60 + a) + sin (60 - a) = \/3 cos a. 

11. cos 5x + cos 9z = 2 cos 7x cos 2x. 
._ sin 7x sin 



= tan 



sin 33 + sin 3 



sin A + sin 



16. cos 20 - sin 10 - sin 50 = 0. 

17. sin (60 + x) - sin x = sin (60 - x). 

18. cos (30 + y) - cos (30 - y) = - sin y. 

19. cos (x + 45) + cos (x - 45) = \/2 cos x. 

20. cos (Q -h 45) 4- sin (Q - 45) = 0. 
01 sin A + sin B 



22. cos 3a cos la = 2 sin 5a sin 2a. 

^ sin 5x sin 2x x 7x 

23. TS ----- E~ = cot -?r- 
cos 2x cos 5x 2 

24. sin + sin 20 + sin 30 = sin 20(1 + 2 cos 0). 

25. cos + cos 20 + cos 30 = cos 20(1 + 2 cos 0). 

26. Express sin x + cos y as a product. 

27. Express sin x cos y as a product. 

28. Show that ~ + tan <* + ^ tan (x - y) = 0. 



136 GENERAL FORMULAS [CHAP. VI 

29. Express as a product, sin a + sin 3a + sin 5a + sin 7<*. 

p cos 5# cos 3x cos 2rc cos 4z __ sin a; ___ 
ov sin 5# sin 3x sin 4# sin 2x cos 4# cos 3o; "~ ' 

31. Prove sin a + sin 2a + sin 3a + sin 4a = 4 cos |a cos a sin fa. 

32. Prove sin a + sin 3a + sin 5a = .- - . 

sin ot 

sin (a + 0) - 2 sin a + sin (a - 0) 

33. Prove- / r-\ o -- r~ ~~? --- "^\ = tana. 

cos (a + p) 2 cos a + cos (a p) 

34. If A + B + C = 180, prove that 

(a) cos (A + B - C) = -cos 2(7. 

ABC 
(6) sin A + sin B sin C = 4 sin -75- sin -$ cos o". 

(c) sin 2A + sin 2# + sin 2(7 = 4 sin A sin 7? sin (7. 

(d) tan A cot B = sec A esc B cos (7. 

35. Prove (cos a + cos 0) 2 + (sin a + sin 0) 2 = 4 cos 2 |(a - 0). 

68. MISCELLANEOUS EXERCISES 

1. (a) Show that the value of sin 20 is less than the value of 2 sin 6 
for all values of B between and 90. 

(6) Show that the value of the fraction ,7 . /, decreases from 1 to 

z mn u 

as 6 increases from to 90. 



2. Given cot a = ^ and cos ft /g-j find the value of each of the 
following if a and /? each terminate in the third quadrant: 

(a) cos (a - 0). (c) sin (0 - a). (e) cot (a - 0). 

(b) tan (a + 0). (d) cot (a + 0). (/) tan (0 - a). 

3. If cos a \ and sin = |, and if a is in the fourth and in 
the third quadrant show that 

(a) sin (a + 0) = +^; cos (a + 0) = -f|; 

tan (a + 0) = -A; 

(b) sin (a - 0) = +1; cos (a - 0) = 0; tan (a - 0) = . 

4. Prove that sin 180 = and cos 180 = 1, using the functions of 
120 and 60. 

6. Find tan (x + y) and tan (x y), having given tan x = and 
tan y = T- 
Verify each of the following: 



58] 
7. 
8. 
9. 

10. tan x + tari y = - 



cot (y - 45) = 
cot (B + 210) 



MISCELLANEOUS EXERCISES 
1 + cot y 



137 



1 cot y' 
. \/3 cot - 1 
cot B + \/3 



sin (x y) tan x tan y' 
sin (x + y) 



cos a; cos y 

+ tan A 

= tan 



tan (0 - 

1 tan (0 <t>) tan 

12. tan (45 + x) - tan (45 - x) = 2 tan 2x. 

13. tan (45 + C) + tan (45 - C) = 2 sec 2C. 

. _ 2 tan x 

sin 2x = q r 



14. 

16. cos 2x = 

16. 1 "*" Kln 
17. 



1 sin 



1 + tan 2 x 

!__ Jan 2 x 
1 + tan 2 x' 



_ /tan cc 
~ Vtan x 



-M) 1 - 



sin 



cos (x 7/) _ 1 + tan x tan y 
cos (x + y} > ~ 1 tan x tan y* 

. . . n sin (A - B) 

tan yl tan B A 

cos A cos J5 



19. 

20. cot x + cot y = 
21. 



sn 



+ y) 



sm x sin y 



, rn0 , v cos A + \/3 sin A 
cos (60 A) = 



cos x sn a; 



22. cos (x - 815) = 

23. cos 5a cos 4 + sin 5a sin 4a = cos a. 

24. sin (x + 75) cos (x - 75) - cos (x + 75) sin (x - 75) = |. 

26. cos (2x + y) cos (x + 2y) + sin (2x + y) sin (x + 2y) 

= cos x cos y + sin x sin y. 

26. sin (x + y) sin (x y) sin 2 x sin 2 y. 

27. cos (x y + z) = cos x cos ?/ cos z + cos x sin y sin 2 

sin x cos ?/ sin z + sin x sin y cos 2. 

28. sin (30 + x) sin (30 - x) = i(cos 2x - 2 sin 2 x). 



138 GENERAL FORMULAS [CHAP. VI 

29. sin (A + B) sin (A - B) = cos 2 B - cos 2 A. 

30. ^sin I + cos |V = 1 + sin s. 



2 sin (45 + ^~2~^) cos ^45 - ^~^\ = cos y + sin x. 



32. 



/ 

33. 1 + tan z tan = sec a;. 



34. tan K + 2 sin 2 ~ cot a: = sin x. 



1 - sin e 

1 tan 

_ _ 1 + sin x + cos x x 

36. r-r- ,. 7- = cot 

37. 1 +cot 2 ^ = 2 

sin tan x 

tan 2 1 + cot 2 ^ 1,0 
2__ ^ _ __ L~t__52_ s x 

j. o x j. x 2 cos x * 

tan 2 o cot 2 - y 

t & 

39. Give the behavior of tan + 2 sin 2 ^ cot 6 as increases from 

t 

to 90. 

40. Show that the value of tan 2 6(1 + cos 20) + 2 cos 2 is the same 
for all values of 0. 

.. _ sin x + cos x 

41. Prove ; _r~ r ~" = tan 2x + sec 2 #- 

J0 cot (90 + A) 

42. Prove 21 1 = csc ' 

-- T^ cos 3x sin 2x cos 4x sin x 

43. Prove ^ ^ ^7 / 3 = ~~ cot 2x> 

44. Prove 4 sin x sin (60 - x) sin (60 + x) = sin 3x. 

45. Find cos 6ct in terms of sin a. 
Verify each of the following: 

46. sin 6 x + cos 6 x = sin 4 x + cos 4 x sin 2 x cos 2 ~. 



58] MISCELLANEOUS EXERCISES 139 

47. sin (x + y z) + sin (x + z y) + sin (y + 2 ) 

= sin (x + y + z) + 4 sin x sin y sin z. 

48. cos x sin (?/ z) + cos y sin (z x) + cos 2 sin (x y) = 0. 

49. sin x cos (y + 2) sin y cos (x + z) = sin (x y) cos 2. 
60. 14 sin 4 x 2 sin 2 x cos 2x = cos 2x. 

51. If a + ft + 7 = 180, prove that 

(a) sin 2 a -f sin 2 ft sin 2 y = 2 sin a sin cos 7. 

a 8 8 y y OL 

(b) tan 2 tan ~ + tan ^ tan 2 + tan 75 tan = 1. 

a ft y a ft y 

(c) cot 2 + cot K + cot TJ = cot 2 cot K cot g* 

52. Prove cos (x + y z) + cos (?/ + z x) + cos (z + x y) 

+ cos (x + y + z) =4 cos x cos ?/ cos 2. 
3) Prove cos (x + y) cos (x ?/) + sin (# + z) sin (?/ z) 

cos (x + z) cos (x 2) ^ 0. 



CHAPTER VII 
IMPORTANT FORMULAS RELATING TO TRIANGLES ' 

69. Law of sines. The object of this chapter is to develop 
important formulas that are useful in solving rectilinear figures 
and to indicate how they are applied. 

In any triangle such as ABC of Fig. l(a), A, B, and C repre- 
sent the angles, and a, 6, and c represent, respectively, the lengths 





Fm. 1. 



(6) 



of the sides opposite these angles. Figure l(a) represents a 
triangle all angles of which are acute; Fig. 1(6), a triangle con- 
taining an obtuse angle. In each figure the line DB is perpendic- 
ular to A C or A C produced. In either figure 



DB . A 

= sin A. 

c 



or 



DB = c sin A. 



(1) 



In Fig. l(a), DB/a = sin C and, in Fig. 1(6), DB/a = sin (180 
C) = sin C. In either case 



DB = a sin C. 



(2) 



Equating the value of DB from (1) to the value of DB from (2) 
and dividing the result by sin A sin C, we obtain 



a 



sin A sin C 



(3) 



Similarly by drawing a perpendicular from C to the opposite 
side of the triangle and reasoning as above, we obtain 

140 



59] 



LAW OF SINES 



a 



sin A sin 
Equations (3) and (4) may be combined in the equations 

a _ b _ c 
sin A ~~ sin B ~~ sin C* 



141 
(4) 

(5) 



The equations (5) are referred to as the law of sines. This law 
may be stated as follows: The sides of a triangle are proportional 
to the sines of the opposite angles. 

Example. Express all line segments of Fig. 2(a) in terms of 
the given parts. 





Fm. 2. 



Solution. Compute the angles of Fig. 2(a) and represent the 
unknown sides by letters; this gives us Fig. 2(6). Attending to 
triangle I, we think: x over sine of angle opposite (75) equals 14 
over sine of angle opposite (35), and write 

14 



x 
sin 75 



or 



sin 35 
Again from triangle I, we write 



y_ 

sin 70 



14 



5 ' r 



x = 14 sin 75 esc 36. (a) 



y = 14 sin 70 esc 35. (b) 



From triangle II, we write 

p = y 

sin 30 sin 45 



or 



sin 30 

p = y - j^o> z 
* sin 45 




sin 45 



(c) 

(d) 



142 



IMPORTANT FORMULAS 



[CHAP. VII 



Replacing y in (d) by its value from (6) and simplifying slightly, 
we obtain 

p = 14 sin 70 esc 35 sin 30 esc 45. 
z = 14 sin 70 esc 35 sin 105 esc 45. 

EXERCISES 

1. Find x and y in radical form from Fig. 3 and also from Fig. 4. 



12 





FIG. 3. 



y 

FIG. 4. 



2. Express x and y in each of Figs. 5 to 8 in terms of the given parts. 





FIG. 5. 



y 

FIG. 6. 




-J 




,60 



FIG. 7. 



x 

FIG. 8. 




3. Find x t y, z, and p of Fig. 9 in terms 
of the given angles. 



FIG. 9. 



59] 



LAW OF SINES 



143 



4. Find sin B where B is defined 
by Fig. 10. Also find the value of 
x in terms of B and the given parts. 

6. Find the area of the triangle 
of Fig. 10 in terms of B and given 
parts. 



6. Express the lines x 
and y in Figs. 11 and 12 in 
terms of a and the given 
angles. 



485 



7. Express the lengths represented by \ 
x, y, z, and w of Fig. 13 in terms of the j 
given parts. 




FIG. 13. 



8. Use Fig. 14 to prove that 



sin A sin B sin C 




9. Show that sin (45 - a) 
where a is defined by Fig. 15. 



sin 85 



FIG. 14. 



48. 




B 



D 
FIQ. 16. 



144 



IMPORTANT FORMULAS 



[CHAP. VII 




10. Express all segments in Fig. 16 in 
terms of a, a, and |8 and then show that 

DT = 17** 

U\j fL\s 



11. If AB = BC in Fig. 17, prove that 

_ sin 40 - sin 30 cos 70 
cota " sin 30 sin 70 " 



60. The law of tangents. Mollweide's equations. The equa- 
tions referred to in the title of this article are easily deduced 
from the law of sines. The law of tangents, the proof of which 
follows directly, is used to solve a triangle when two sides and the 
included angle are given. Mollweide's equations are excellent 
equations for checking purposes. 

From the law of sines, we have 



a _ sin A 
b sin B 

Subtracting 1 from each side of (6), we have 

a ^ sin A . a b sin A sin B 



or 



1 A TS v/A i 

6 sin B ' b 

Adding 1 to each side of (6), we have 



sin B 



a 
b 



sinA 



or 



a + b sin A + sin B 



(6) 



(7) 



(8) 



n I * Vi i. r 

sm B b sin B 

By dividing (7) and (8) member by member, we obtain 

a b __ sin .4 sin_B 
a + 6 ~~ sin A + sin B 

Transforming the right-hand member of this equation by means 
of the formulas of 57, we obtain 



60] LAW OF TANGENTS. MOLLWEIDE'S EQUATIONS 145 

sin A - sin B = 2 cos %(A + B) sin %(A - B) 
sin A + sin B ~~ 2 sin (A + B) cos %(A - B) 

The right-hand member reduces to 

tan %(A - B) -f- tan (A + B). 

g ~ b = tan ^Q4 - B) 
" ' a + b tan |(4 + B) ( ' 

Another formula may be obtained by replacing a by c and 
A by C in (9) and a third, by replacing 6 by c and B by C in (9). 

When b > a, both sides of (9) are negative. In this case it 
is convenient to write the formula in the form 

% ' *>-<* = tan_i(ir-jl) 

6 + tan 1/D ' " N> U ' 



so that both members are positive. 

The formulas often called Mollweide's equations are derived 
as follows: 

From the law of sines, we have 

a- sin A , 6 sin B /<<x 

'c^sTFc' and c^iSTc' (1W 

Adding equations (11) member by member, we obtain 
a + b sin A + sin B 



sin C 



(12) 



Transforming the right-hand member of this equation by means 
of formula (18) of 56 and formula (33) of 57, we obtain 



a + b = 2 sin fr(A + B) cos j(A - B) 
c 2 sin ^C cos ^C ' 

Since A + B = 180 - C, 

sin (A + 5) = sin (180 - C) = cos %C. 
Hence Mollweide's first equation may be written in the form 



sin C * ; 



146 



IMPORTANT FORMULAS 



[CHAP. VII 



Mollweide's second equation, 

a-b _ sin (A - B) 
~~~ 



is derived in a similar manner. 

61. The law of cosines. In the triangles of Fig. 18 denote 

the angles by A, B, and C, and the sides opposite these angles 

by a, 6, and c, respectively. Draw the perpendicular p from 

C 




c 




-cto- 



D -K - c 3 

(a) (6) 

FIG. 18. 

one of the vertices C of the triangle to the opposite side c, Fig. 
18(a), or c produced, Fig. 18(6). In either figure 

AD = 6 cos A. (16) 

In Fig. 18 (a) 

DB = c AD = c 6 cos A, 
and in Fig. 18(6) 

BD = AD - A5 = 6 cos A - c. . (17) 

Since (c 6 cos A) 2 = (6 cos A c) 2 , we have for each triangle 

6 2 6 2 cos 2 A = p 2 = a 2 (c 6 cos A) 2 . 
Simplifying and solving for a 2 , we obtain 

a* = b* + c 2 - 2bc cos A. (18) 

Similarly, by drawing perpendiculars from A and Z? to the 
cmposite sides or the opposite sides produced, we obtain 

b 2 = a 2 + c 2 2ac cos J 
c 2 = a 2 + b 2 2ab cos 

The law of cosines embodied in equations (18) and (19) may 
be stated as follows: The square of any side of a plane triangle 
is equal to the sum of the squares of the other two sides diminished 
by twice the product of those two sides and the cosine of their included 
angle. 



61] 



THE LAW OF COSINES 



147 



The law of sines, the law of cosines, and the law of tangents 
will be used in the next chapter to compute parts of rectilinear 
figures. Here we shall use 'them to write expressions for lengths 
of line segments of rectilinear figures and to write identities. 

Example 1. Write several equations relating to Fig. 19. 
Solution. From the law of sines, we 
have 



x = V 
sin 40 sin 65 



20 



sin 75 



Substituting a = 20, A = 75, 6 = x, 
c = y in (18), we obtain 




20 
FKJ. 19. 



20 2 = x 2 + y 2 - 2jcy cos 75. 
Substituting a = 20, A = 75, b = x, B = 40 in (9), we obtain 

20 - x _ tan j(75 - 10) = tan (1730') 
20 + "x " tan (75 + 10) " Ian (573(V)' 

Substituting a = 20, .1 = 75, b = j, B = 40, c = y, C = 65 
in (14), wo obtain 



20 



+ y _ cos ^(75 - 40) _ cos (1730 ; ) 
V "" sin 4(65) " " * (3230 / )' 



Example 2. Express the lino segments j, y, z, and w of Fig. 
2()(a) in terms of the given parts, and write an identity based on 
these results. 




FIG. 20. 



148 



IMPORTANT FORMULAS 



[CHAP. VII 



Solution. First we devise Fig. 20(6). Applying the law of 
sines to triangle ABD of Fig. 20(6) we obtain 



1 



sin (a + 0) sin 



= esc 0, 



y 



sin a 



esc 0, 



(a) 



or 



x = sin (a + 0) esc 0, y = sin a esc 0. 
Applying the law of sines to triangle DBC, we obtain 



sin (90 - 0) sin (a + 



sin (90 - a) 



Replacing y by sin a esc 0, solving for z and w, and simplifying 
slightly, we have 

w = tan a cot 0, z = sin (a + ft) tan a esc 0. (d) 
Applying the law of cosines to triangle BDC, we obtain 

cos (a + #). (e) 

Replacing y, z, and w by their values from (b) and (d), we obtain 

sin 2 (a + ft) tan 2 a esc 2 = sin 2 a esc 2 + tan 2 a cot 2 

2 sin a esc tan a cot cos (a + 0). 

EXERCISES 



1. Use the law of cosines to find x in 
Fig. 21; then express sin A and sin # in 
terms of x. 




6 
FIG. 21. 




2. In Fig. 22 find tan 
using formula (9) in 60. 



- 5) by 



FIG. 22. 



62] 



MISCELLANEOUS EXERCISES 



149 




3. In each of Figs. 23 and 
24 use the law of cosines to 
find x. Then express sin A 
and sin B in terms of x. A 



4. In each of Figs. 25 and 26 find tan 1(A B) by using formula 
(9) in 60. 




6 = 3 



a =4 



c=5 
FIG. 25. 




62. MISCELLANEOUS EXERCISES 

In the following exercises check each identity by substituting one or 
more of such angles as 0, 30, 45, 120, 240, 270, etc., for the unknown 
angles involved. 



1. Use the law of cosines to find the 
value of x in Fig. 27. 

2. Find the value of tan ?(A - B) 
where A and B are defined by Fig. 27. 

3. Find an expression for the area of 
the triangle in Fig. 27. 




33 
Fio. 27. 



4. Write equations applying to Fig. 28 by 
using each of the following: law of sines, law of 
cosines, law of tangents, Mollweide's equations. 




Fia. 28. 



150 



IMPORTANT FORMULAS 



[CHAP. VII 





6. Find an expression for the area 
of the triangle in Fig. 29 in terms of c, 
A, and B. 

Hint. First find x and then h. 



6. Find the value of cos A where A is 
defined by Fig. 30. 



12 
u. 30. 




7. (a) From Fig. 31 find a 
formula for h in terms of the given 
parts. 

(6) Using the formula found 
in (a), compute h. 



8. Using Fig. 32, express h in terms of 
m, x, y, z, w. 



9. Find the length of all line segments of 
Fig. 33 in terms of the given parts. 



FIG. 33. 



$62] 



MISCELLANEOUS EXERCISES 



151 



10. Draw the altitude to the side 
lettered x in Fig. 34 and find its 
length in terms of 6 and <p; then 
write a formula for the area of the 
triangle. Check this formula by us- 
ing the values = 90, <p = 45. 

11. In Fig. 35 trihedral angle has 
the face angles a, 6, c, and trihedral 
angle (J has the face angles C, 90, 90. 
Express the length of each line segment 
in terms of a, fc, r, then find and equate 
two line values of DE, and simplify to 
obtain cos c = cos a cos b + sin a sin b 
cos C. 




D 



FKI. 35. 



12. From the law of cosines derive algebraically the law of sines. 
Hint. Find cos A in terms of a, /;, and r; then find (sin 2 A) /a 1 = 



13. 0-ABC in Fig. 36 represents a pyramid. 
Find the length of each edge in terms of a, 
0, 7, 0> and <p. 




FIG. 3ft. 



CHAPTER VIII 
OBLIQUE TRIANGLES 

63. Introduction. In this chapter wo shall develop formulas 
and exhibit plans of calculation for the solution of oblique 
triangles. 

When the length of a side and two other parts of a triangle 
are known, the remaining parts can generally be found. The 
four cases that arise in the solutions of oblique triangles are 
referred to as 

Case I. Given one side and two angles. 

Case II. Given two sides and an angle opposite one of them. 

Case III. Given two sides and the included angle. 

Case IV. Given three sides. 

All triangles can be solved by means of the law of sines, the 
law of cosines, and the law of tangents. However, formulas 
especially adapted to logarithmic computation will be developed 
to solve triangles classified under Case IV. Although any 
formula not used in the solution of a triangle may be used as a 
check formula, Mollweide's equations are particularly desirable 
check formulas because they contain all six parts of the triangle 
and are well adapted to logarithmic computation. A single 
setting of the slide rule will serve to check, within its range of 
accuracy, the solution of any triangle. 

For convenience of reference wo repeat here the slide-rule 
setting for applying the law of sines to solve a triangle: 

Rule A. To apply the law of sines for solving a triangle, 
push the hairline to any known side on D, 
draw under the hairline the opposite known angle on S; 
push the hairline to any other side on D, 
read at the hairline the angle opposite on S; 
push the hairline to any other known angle on /S, 
read at the hairline the side opposite on D. 
152 



65] CASE I. GIVEN ONE SIDE AND TWO ANGLES 153 

64. Form for computation by logarithms to be used in the 
solution of oblique triangles. The student should now recall 
the forms and the general method of procedure used in the solu- 
tion of right triangles by logarithms. When oblique triangles 
are solved, a similar method will be used. This method may be 
summarized as follows: 

a. Draw a figure of the triangle to be solved, lettering it in the 
conventional way. Encircle the given parts. 

b. Write the formulas to be used in the solution. 

c. Make a complete form for the computation before looking up 
any logarithms. 

d. Fill in the form. 

65. Case I. Given one side and two angles. 

Example. Given a = 24.31, A = 4518', and B = 2211' 
(see Fig. 1). Find 6, c, and C. 




Solution. Since A + B + C = 180, 

C = 180 - (4518 / + 2211 / ) = 11231'. 

To find ?;, choose the formula from the law of sines which contains 
b and three known parts. Solve this formula by algebra for 6, 
to obtain 



Similarly, 



, a sin B . n A f , 

o = -. 7 = a sin B esc A. (a) 

sm A 



a sin C . ~ A /, N 

c __ = a sm C esc A. (b) 

sm A 



The solution for the unknown parts in (a) and (6) and the check 
by Mollweide's equation (14) 60 are displayed below. The 
letter in parenthesis above each column refers to the formula 
associated with the column. 



154 


OBLIQUE TRIANGLES 






[CHAP. 


VIII 










(a) 




(b) 






a 


= 


24. 


31 


log a 


= 1.38578 


log 


a 


= 1 


38578 




A 


= 


45 


18' 


7 % A 


= 0.14825 


/ CSC* 


A 


= 


14825 




B 


= 


22 


11' 


1 sin B 


= 9.57700- 10 












b 


= 


12. 


913 log b 


= 1.11103 












C 


= 


112 


31' 




/ sin 


C 


= 9 


96556 


- 10 


c 


= 


31. 


593 


log c 


= 1 


.49959 





Check. For convenience of computation, we write Mollwcide's 
equation (14) of 60 



in the form 



a + b 



- -sn 



- B) 



sn 



sec 



a + b = 37.223 
c = 31.593 
%C = 5615 ; 30" 

- E) = 1133'30" 
1 



sec 



- B) = 1. 

log (a + b) = 1.57081 

colog c = 8.50041 - 10 
Zsin^C = 9.91989 - 10 
- B) = 0.00890 




24.31 
Fio. 2. 



log 1 = 0.00001 

To solve the triangle by means of 
the slide rule, we first find C = 
11231' from the relation A + 
B + C = 180 and then use Rule A 
of 63. Hence, construct the tri- 
angle shown in Fig. 2, and 



push hairline to 24.31 on D, 

draw 4518' of S under the hairline, 

push hairline to 2211' on S, 

at the hairline read b = 12.91, 

push hairline to 6729 ; (= 180 - 112 31') on 8, 

at the hairline read c = 31.6. 

* Note that I is used in these forms to abbreviate the word log. If your 
tables of logarithms of trigonometric functions do not give the values of the 
logarithms of the secant and cosecant, in the above form write colog cos 
for I sec and colog sin for Z esc. 



65] 



CASE I. GIVEN ONE SIDE AND TWO ANGLES 



155 



EXERCISES 

Solve the following triangles: 

1. A = 5428', 3. A = 6456'18", 

B = 1038', B = 4729 / ll", 

a = 3.695. c = 913.45. 



2. B = 3812'48", 
C = 60, 
a = 7012.6. 



A = 4723'18", 
C = 7016'49' / , 
c = 227.22. 



5. A = 7113'30", 
B - 4034'15". 

c = 236.53. 

6. A = 2532'35", 
B = 13313'5", 
a = 411.41. 




7. A line AB along one bank of a stream is 562 ft. long, and C is a 
point on the opposite bank. The angle B A C is 53 18', arid the angle 
ARC is 4836'. Find the width of the stream. 

8. A vertical plane contains a 132-ft. hillside tunnel sloping down- 
ward at 14 with the horizontal and cuts the hillside in a line sloping 
upwards at 18. What is the vertical distance from the bottom of the 
tunnel to the surface of the hill? 



9. Prove that the area A' of 
triangle ARC in Fig. 3 is given by 

_ b 2 sin A sin C 
A = 2~sfn (.4 + C)' 

Hint. First find c in terms of 
encircled parts; then find h and 
use the formula K = -^cA. 

10. Use the formula in Exercise 9 to find the area of the triangle in 
(a) Exercise 1 ; (b) Exercise (>. 

11. A shore station at point A is 5280 ft. from another at point B. 
Find the distance from each of the shore stations to an enemy ship at 
point (7 if angle ABC is 8337'* and angle BAC is 85 < T. 

12. A surveyor running a line due east reached the edge of a swamp. 
He then ran a line 2000 ft. in a direction S. 47 E., and from the point 
thus reached he ran a line in the direction N. 52 20' E. How far had 
he continued on this latter line when he reached a point on the original 
line extended? 

13. A building 75.2 ft. high stands at the upper end of a street that 
slopes down at an angle of 652' with the horizontal. How far down the 
street from the building is a point at which the angle of elevation of the 
top of the building is 1358'? 

14. From the top of a hill the angles of depression of the top and the 
bottom of a building 42.5 ft. high are observed to be 36 and 43, 



156 



OBLIQUE TRIANGLES 



[CHAP. VIII 



respectively. Find the height of the hill if the building is at the foot 
of the hill. 

66. Case II. Given two sides and the angle opposite one of 
them. In this case, as in Case I, the triangle is solved by means 
of the law of sines and the relation A + B + C = 180. The 
result may be checked by means of Mollweide's equations. 
However, this case needs further discussion, for in one instance 
an ambiguity exists. 

Ambiguous case. When the side opposite the given angle is 
less than the other given side, there are three possibilities: no 
solution, one solution, or two solutions. Let us investigate the 
situation in detail. 

Let A, a, and b of Figs. 4, 5, 6 be the given parts in which 
a < 6. The perpendicular from C to side c is b sin A. 

a. If, in Fig. 4, a < b sin A, side a is too short to reach side c. 
Hence there is no solution. 

C 




a<b sin A 



a-b sin A 



c 

Fm. 4. 



B 



Henoe 




> c 2 J-- ' 

l< c t - 



FIG. 6. 



6. If, in Fig. 5, a = b sin A, side a just reaches side c. 
there is one solution, a right triangle. 

c. If, in Fig. 6, a > b sin A, 
there are two solutions. In prac- 
tice this is the most probable 
condition. Notice that BI and 
#2 are supplementary angles. 

These results may be sum- 
marized thus : If in triangle ABC, 
a < b, we have no solution when a < b sin A ; one solution when 
a = b sin A ; two solutions when a > b sin A . 

In the ambiguous case it is not necessary to determine the 
number of solutions in the foregoing manner before proceeding 
to solve the triangle, for we shall discover the nature of the 
situation as soon as we have added the first column of logarithms 
in the solution. Hence proceed with the computation, and 
when log sin B has been found observe that 



CASE II 



157 



(a) if log sin B > 0, then sin B > 1, and there is no solution; 

(b) if log sin 5 = 0, then sin B = 1 and there is one solution, a 

right triangle ; 

(c) if log sin B < 0, then sin B < 1, and there are two solutions. 

Hence in Case II the procedure is as follows: 

a. Determine whether the ambiguous case exists by noting 
whether the side opposite the given angle is less than the side 
adjacent to the given angle (a < b). 

b. Proceed with the computation and if the ambiguous case is 
involved expect two solutions, but keep in mind that there may 
be no solution or one solution. 

Example 1. Given a = 67.528, b = 56.827, and A = 79 
15'20" (see Fig. 7). Find c, J5, and C. 

Solution. By inspection it is observed 
that a > b. Hence this is not the ambigu- 
ous case. 

To find B, from the law of sines choose 
the formula containing B and the three 
known parts. Solve this formula for B to 
obtain 

b sin A , x 



sin B = 

After finding B from (a), determine C from the relation 
A + B + C = 180. 

Then write the law of sines involving c, C, and the knowns 
and A to obtain 

a sin C 




c = 



= a sin C esc A. 



(b) 



sin A 

The solution is displayed in the following form. The letter in 
parenthesis above each column refers to the formula associated 
with the column. 



(a) 


(b) 






a 


= 67.528 


colog 


a 


= 8 


17051 - 


10 


log 


a 


= 1. 


82949 




b 


= 56.827 


log 


b 


= 1 


75456 














A 


= 7915 / 20" 


/sin 


A 


= 9 


99232 - 


10 


I csc A 


= 


00768 




B 


= 5546'8" I sin 


B 


= 9 


.91739 - 


10 












C 


= 4468'32" 










I sin 


C 


= 9 


84930 


- 10 


c 


= 48.581 








log 


c 


= 1 


68647 





158 



OBLIQUE TRIANGLES 



[CHAP. VIII 



The results should be checked by means of one of Mollweide's 
equations, as in Case I. One setting of the slide rule serves to 
check the results. 

To solve Example 1 by means of the slide rule, set the pro- 
portion 

67.5 = 56.8 = c_ 

sin 7915' ~~ sin "5 ~ sin C 

on the rule, and read B = 6546'. From the relation A + B + 
C = 180, get C = 45; then on the slide rule read c = 48.6. 

Example 2. Given a = 9.467, b = 14.433, and A = 1114'18" 

(see Fig. 8). Find c, J5, and 
(b) _^-- ^^_ffl C. 

Solution. By inspection it 
is observed that a < b. 
Honce this is the ambiguous 

case. When log sin B has been computed, we shall determine 
the number of solutions. The formulas, obtained as in Example 
1, are 




sin B 



b sin A 

> 

a 



C = 180 - (A + B), 
a sin C 



c = 



sin A 



= a sin C esc A. 



(a) 
(b) 



The solution is displayed in the following form: 



(a) 


W 


(6) 




a 


=3 


9 467 


col OK a 


= 9 


02379 - 


10 


loga 


= 97621 


log a 


- 97621 




6 


= 


14 433 


IOR& 


- 1 


15936 














A 


- 


1114'18" 


I sin A 


- 9 


28979 - 


10 


Jcsc A 


= 0.71021 


1 CMC A 


= 71021 




Bi 





1717 / 6 // 


iBinBi 


=- 9 


47294 - 


10 












82 


_ 


16242'04" 




















Ci 


_ 


15128'36" 










I sin C\ 


= 9.67899 - 10 








d 



















I sin Cz 


= 9 02259 - 


10 




. 


23.196 








logci 


- 1.36541 








ci 


=3 


5.1169 










logC2 = 


= 70901 





Since log sin B from the first column was found to be negative, 
we concluded that there were two solutions. Since sin B is 
positive in both the first and the second quadrants, we obtained 
two supplementary angles BI and B z from log sin B. 



66] 



CASE II 



159 



One of Mollweide's equations should be employed to check 
the results. It is interesting to check the results of both solu- 
tions by a single setting of the slide rule. 

To solve the triangle of Example 2 by means of the slide 
rule, use the same general line of argument applied in the log- 
arithmic solution, but employ Rule (A) of 63 for the computa- 
tion. Hence draw Fig. 9 and 

C, 

;180-Ci 




FIG. 9. 

push hairline to 947 on D, 

draw 1114' of S under hairline, 

push hairline to 14.43 on D 9 * 

at the hairline read BI = IT !?' on S; 

push hairline to 180 - Ci = 2831 / on ti, 

at the hairline read c\ = 23.2 on /); 

compute C 2 = #1 - 1114' = 63', 

push hairline to 63' on S, 

at the hairline read c 2 = 5.12 on D. 

Example 3. Given a = 96.55, 6 = 124.98, and A = 5034'51" 
(see Fig. 10). Find c, B, and C. 

Solution. Upon observing that a < b, we know that this is 
the ambiguous case. The number of solu- 
tions will be determined from log sin B. 
The formulas, obtained as in Example 1, are 

. n b sin A 

sin B = > 

a 

C = 180 - (A + B), 
a sin C 



c = 



sin A 



= a sin C esc A. 




* Occasionally it will be necessary to use the following rule: when a 
number is to be read on the 1) scale opposite a number on the slide and 
cannot be read because the slide projects beyond the body of the rule, push 



160 



OBLIQUE TRIANGLES 



[CHAP. VIII 



The solution is displayed in the following form: 



a = 96.55 
6 = 124.98 
A = 5034'51'' 
B = 9000'00" 
C = 3926'9" 
c = 79.360 


(a) 
colog a = 8.01525 - 10 
log b = 2.09684 
I sin A = 9.88791 - 10 


(W 
log a = 1.98475 

Jcsc A = 0.11209 
Z sin C = 9.80276 


- 10 


I sin B = 0.00000 


log c = 1.89960 



While computing, we found that log sin B = 0. Therefore 
sin B = 1, B = 90, and there is one solution. 

The computation should be checked by one of Mollwcide's 
equations. 



EXERCISES 



Solve the following triangles: 

1. a = 309, 
b = 360, 

A = 2114 / 25". 

2. b = 316, 
c = 360, 

B = 2116'44". 

3. A = 4113', 

a = 77.04, 
b = 91.06. 

4. b = 115.97, 
c = 139.06, 

J5 = 4311'32 // . 
6. a = 294, 

b = 189, 

A = 6732'. 
6. b = 71.818, 

c = 78.493, 

B = 6612'10". 



13. It is desired to measure the distance AB between two points on 
opposite sides of a lake. A point C, easily accessible to both A and B, 



7. a = 48.134, 
6 = 35.826, 

A = 3624'0". 

8. a = 32.239, 
b = 50.204, 

A = 3218 / 30 // . 

9. a = 4.2356, 
6 = 5.1234, 

A = 5418'0". 

10. b = 216.45, 
c = 177.01, 

C = 3536'20". 

11. a = 341.91, 
b = 745.91, 

A = 4335'39". 

12. a = 95.21, 
b = 126.4, 

A = S 



the hairline to the index of the C scale inside the body and draw the other 
index of the C scale under the hairline. The desired reading can then be 
made. 



67] CASE III. GIVEN TWO SIDES AND INCLUDED ANGLE 161 

is chosen. It is found that AC = 8461 and BC = 10,246. At A the 
angle B-AC is found to be 2633'. Find the distance AB. 

14. Two wires are run from the same point on the vertical edge of a 
building to a level courtyard below. One wire is 42.45 ft. long and 
makes an angle of 58 with the horizontal. The other wire is 48.60 ft. 
long and lies in the same vertical plane with the first but on the oppo- 
site side of the edge. Find the inclination of the second wire to the 
yard and the distance between anchor points. 

16. The distance from a point A to a point C cannot be measured 
directly but is estimated to be about T mile. From a point B, BA = 
7201.5 ft., and BC = 6180.3 ft. Angle BAG is found to be 4114'25". 
Find the distance AC. 

67. Case III. Given two sides and the included angle. When 
two sides and the included angle are the given parts, the triangle 
can be solved by means of the law of tangents and the law of 
sines. The law of tangents gives the angles opposite the given 
sides, and the law of sines can then be used to find the third 
side. The result may be checked by means of Mollweide's 
equations. 

Example 1. Given c = 1.0398, a = 6.7517, and B = 1279'18" 
(see Fig. 11). Find A, C, 
and b. 

Solution. From the relation ^ 
A + B + C = 180, we have 
A + C = 180 - 5, or 

%(A + C) = (180 - 1279'18") = 2625'21 /7 . 
From the law of tangents, (see 60) we have 

tan (A - C) = ? ,\ tan (A + C), (a) 




and from the law of sines 

, a sin B . D , * /1A 

b = . 7- = a sm B esc A.* (o) 

sm A ^ ' 

* In this case one of Mollweide's equations may be used to find the 
unknown side and the other as a check. 



162 OBLIQUE TRIANGLE [CHAP. VIII 

The solution is displayed in the following form: 



= 1279'18" 
o=6.7517 
c = 1.0398 
a-c = 5.7119 
7.7915 



log (a -c) =0.75678 
colog (a +c) =9. 10838 -10 
I tan ^ (A +C) =9.69626 - 10 



-C) = 200'54" I tan $(A -C) =9.56142-10 
A=4626'16" 
C = 624'27" 
6 = 7.4262 



/sin 5 = 9.90146-10 
log a = 0.82941 



I esc A =0.1 3989 



log b - 0.87076 



The following solution will illustrate the method of using the 1 
slide rule to solve a triangle when two of its sides and the included 
angle are known. 

Example 2. Solve the triangle in which b = 28.7, c = lf>.2, 
A = 47. 

Solution. In Fig. 12 draw line CD perpendicular to AH, and 
solve the right triangle ACD. Knowing jc, get z = 4f>.2 j-. 




B 



!<- - -S-CM5.2 --4 

FIG. 12. 

Then, knowing the two legs y and z of right triangle D#C, solve it 
by the method of 128. This leads to the following settings: 

set right index of C to 28.7 on D, 
opposite 43 on S read x = 19.6 on Z), 
opposite 47 on S read y = 21 on Z); 
compute z = 45.2 - 19.6 = 25.6, 
set right index of C to 25.6 on Z), 
push hairline to 21 on D, 
at hairline read B = 3922' on T\ 



67] CASK III. Gl YEN TWO SIDES AND INCLUDED ANGLE 163 



draw 3922' of S under the hairline, 
opposite index of C read a = 33.1 on D. 
Evidently angle C = 43 + 90 - 3922' = 9338 ; . 

EXERCISES 

Solve the following triangles: 



6. b = 85.249, 
c = 105.63, 
.1 = 504()'24' ; . 

6. a = 0.59312, 
6 = 0.22734, 
C = 6438'0". 

7. a = 6.2387, 
6 = 2.3475, 
C = 11032 ; . 

8. a = 35.237, 
6 = 18.482, 

C = 11040'30", 



l.o = 17. 
b = 12, 
C = 5917'. 

2. o = 748, 
b = 375, 

C = 6335'30". 

3. 6 = 232.23, 
c = 195.59, 

A = 6113'0". 

4. a = 27.92, 
b = 42.38, 
C = 3940'. 

9. The end A of a boom AB is 
attached to the platform of a 
crane and a cable BC connects 
the end B to a point C on top of 
the crane (see Fig. 13). If AB = 
35 ft., AC = 15 ft., and angle 
CAB = 95, find the length of the 
cable. 

10. From a point 5890 ft. from one end of a lake and 6728 ft. from the 
other end, the lake subtends an angle of 4718 ; . Find the length of the 
lake. 

11. A triangular tract of land is to be enclosed by a fence. The side 
AB = 54.235 ft. ; side CB = 29.483 ft. ; the included angle B is 9540'25". 
Find the amount of fencing needed to enclose the triangular plot. 

12. From the top of a lighthouse 188.6 ft. above sea level, the angle of 
depression of a ship was 530'30", and its compass bearing was 1648'0". 
One hour later the angle of depression was 410 ; 0" and the compass bear- 
ing, 1434'0". Find the distance traveled by the ship and its compass 
course. 

13. Two yachts start from the same place at the same time. Yacht A 
sails at 10 knots on compass course 62. Yacht B sails at 8 knots on 
compass course 135. How far apart are they at the end of 40 min., 
and what is the bearing of yacht B from yacht A ? 




FIG. 13. 



164 OBLIQUE TRIANGLE [CHAP. VIII 

14. Prove that the area K of the triangle shown in Fig. 14 is given by 
K = $ab sin C. 

Use the formula just derived to find the area of the triangle of (a) 
Exercise 1; (6) Exercise 7. 




FIG. 14. 

16. From a mountain peak in a vertical plane through a straight 
tunnel, the angles of depression of its ends are 424i' and 5222', and 
the corresponding distances from the peak to the ends of the tunnel are 
3710 ft. and 4100 ft., respectively. Determine whether the tunnel is 
horizontal and find its length. 

16. From a ship two lighthouses bear N. 40 E. After the ship has 
sailed 15 miles on a course of 135, they bear 10 and 345, respectively. 
Find the distance between them and the distance from the ship in the 
latter position to the more distant lighthouse. 

17. Two men, A and J?, start at the same point on the circumference 
of a circle of radius 900 ft. and walk at the rate of 350 ft. per minute. 
If A walks toward the center of the circle and B walks along the circum- 
ference, find how far apart the two men are at the end of 1 min. 

68. The half -angle formulas. While the law of cosines may be 
used to solve a triangle when the three sides are given, it is not 
convenient to use in logarithmic computation. We shall now 
derive from the law of cosines other formulas that are well 
adapted to logarithmic computation. 

From the first equation of (24) 56, we obtain 

2 sin 8 A = 1 - cos A, (1) 

and from the law of cosines, we have 

, 6 2 + c* - a* , . 

008 A We (2) 



68] THE HALF-ANGLE FORMULAS 165 

Substituting the value of cos A from (2) in (1), we get 



2bc - b 2 - c 2 + a 2 

2bc 
a 2 - (fr 2 - 2bc + c 2 ) 

26c 
a 2 - (b - c) 2 

26c 

_ (a + 6 - c)(a - b + c) , . 

- 2fo (3) 

Let 

a + 6 + c = 2s. (4) 

Subtracting 2a, 26, and 2c from each member of (4), we obtain, 
respectively, 

-a + b + c = 2(s - a), 
a - 6 + c = 2(s - 6), 
a + 5 - c = 2( - c). 

Substituting from the last two of these equations in (3) and 
simplifying slightly, we get 



-^ (5) 

Similarly, 



rinifl-^ "^ % (6) 

and 

/7 " Z 1F E &)- (7) 

Using the second definition (8) of 4 together with (1) above, 
we have 

sin 2 ^ A = hav A. 

From this equation and (5), we easily derive 

hav^ = (g ~ b ^~ c) . (8) 



166 OBLIQUE TRIANGLE [CHAP. VIII 

Similar formulas for hav B and hav C may be obtained from (6) 
and (7). Formula (8) is often used when haversine tables are 
available. 

From the second equation of (24) 56 and (2), we obtain 

h'2 J- /2 __ n '2 

r t> 1 A 1 I W I ^ ** 

2coB'p = l + - 2&- - 

= 2bc + b 2 + c 2 - a 2 

2bc 

(6 + c) 2 - a 2 
26c 

= ( + b + c )(~ a + *> + c) 
2bc~ 

(28)2(8 - a) 

2bc 
Hence 



-fc-- (9) 

Similarly, 

cos P = ~. (10) 

and 



coBK-v-ri (") 

1 sm "M 
since tan j%A > we get by substitution from (5) and 



cos 



tan -L4 = .P , /x , "' (12) 

2 \ *( a) v ; 

Similarly, 
and 



g( /I c) ' (14) 

Formula (12) may be written 



tan *A = -^-^v" <'* ^ -/v_^/. (15) 



69] 

If we let 

we may write 
Similarly 



CASE IV. GIVEN THREE SIDES 



167 



r _ 

r _ 



- c) 



tan 



tan C = 






(16) 

(17) 
(18) 



When calculating the angles of a triangle, the tangents of the 
half angles should he used, since the complete calculation of .4, B, 
C may be performed by taking from the tables only the four 
logarithms log s, log (s a), log (s 6), and log (s c). 

69. Case IV. Given three sides. When the three sides of a 
triangle are given, its solution may be effected by means of the 
half-angle formulas and the results 
checked by means of the relation 
A + B + C = 180. 



Example. Given a = G.8235, b = 
5.2063, and c. = J 
Find A, B, and C. 




where 



1C. 


"-C 


he half-angle formulas arc I0 ' ' 




4- o . 


(a) 


ta " 2 ~ - a 


. B r 

TO T\ _ . . . } 


(6) 


tan 2 s - 6 


* C r 


(c) 


tan 2 s - c 


/(s - o)(s - 6)( - e) r ,, 



11 r is the radius of the circle* inscribed in the triangle. 



168 OBLIQUE TRIANGLE 

The solution is displayed in the following form: 



a- 6.8235 

6= 5.2063 

c= 3.1628 

2= 15.1926 

= 7.5963 

t-o= 0.7728 

-&- 2.3900 

-c= 4.4335 

= 7T963 



colog t*9.11940-10 
log -o-9.88807 -10 
log -6 =0.37840 
log -c =0.64675 



2)logr'0.03262 
log r =0.01631 



(a) 



colog-o=O.I1193 



(6) 



[CHAP. VIII 



(*) 



log r= 0.01631 
0.12824 



col og -6 -9.621 GO- 10 



log r= 0.0 1631 



4657'42" ifl- 
2621'S8" iC- 

-lSOW (CAecJb) 



colog-c- 9.35325 -10 



logr-0.01631 



C- 9.36956 -10 



arithmetic involved in computing s a, s 6, arid s c 
was checked by verifying that their sum was s. 

By means of the law of cosines, we can find by the use of the 
slide rule one of the angles of the triangle. Then, by applying 
the law of sines, we read on the slide rule the other two angles. 

EXERCISES 

Solve the following triangles: 

1. a = 3.41, 6. a = 95.321, 
6 = 2.60, 6 = 113.72, 
c = 1.58. c = 179.84. 

2. a = 111, 6. a = 2.2361, 
6 = 145, b = 2.4495, 
c = 40. c = 2.6458. 

3. a = 14.493, 7. a = 1.4932, 
b = 55.436, b = 2.8711, 
c = 66.913. c = 1.9005. 

4. a = 97.862, 8. a = 529.37, 
b = 105.98, b = 716.49, 
c = 138.72. c = 635.21. 

Use the law of cosines to solve the following triangles: 

9. a = 13, 11. a = 60, 

6 = 11, b - 40, 

c = 9. c = 35. 

10. a = 6. 12. a = 2. 

6 = 7, 6 - 3, 

c 8. c = 4. 



70] 



SUMMARY 



169 




13. Find the largest angle of the triangle whose sides are 13, 14, 16. 

14. To find the width of a river, a point A 
(Fig. 16) is located on one bank and two 
points B and C on the other. A fourth point 
D is located in line with AB, and a fifth point 
E in line with AC. The distances were meas- 
ured as follows: BC = 506 ft., BD = 453 ft., 
BE = 809 ft., CD = 753 ft., CE = 392 ft. 

Find the width of the river. 

FIG. 1G. 

16. Three towns, A, B, and C, are situated so that AB = 23.37 miles, 
BC = 11.84 miles, and AC = 16.29 miles. A road from A to B is 
met at D by a perpendicular road from C. 
latter road and the distance DB. 

16. Derive Heron's formula for 
the area K of a triangle in terms 
of its three sides a, 6, c, and s = 
\(a + b + c), namely: 

K= " 
Hint. 



Find the length of this 





(b) 
Fia. 17. 



/s(s - a)(s - b)(s - c). 

The area of the triangle 
shown in Fig. 17 is K = -$bh = -%cb sin A 
A. Replace sin A by 2 sin I A 
cos ^A, and then use (5) and (9). 

17. Use Heron's formula to find the area of the triangle of (a) Exer- 
cise 1; (6) Exercise 7. 

18. The sides of a triangular field measure 223.6 ft., 244.9 ft., and 
264.6 ft. Find the area of the field. 

70. Summary. A summary of the four cases of oblique tri- 
angles is given below in tabular form. 



Given 


One side and two 
angles 


Two sides and the 
angle opposite 
one of them 


Two sides and the 
included angle 


Three sides 


Using loga- 
rithms, 
solve by 


Law of sines 


Law of sines 


Law of tangents 
and law of sines 


Tangent of half- 
angle formulas 


Using slide 
rule, solve 
by 


Law of sines 


Law of sines 


Dropping a per- 
pendicular 


Law of cosines 
and law of sines 


Check by 


Mollweide's equations 


A + B + C - 
180, and slide 
rule 



170 



OBLIQUE TRIANGLE 



[CHAP. VIII 



71. MISCELLANEOUS EXERCISES 

Solve the following triangles: 

1. o 42.365, 3. a = 412.67, 5. a = 6.342, 
6 = 25.863, A = 5038'50", 6 = 7.295, 
C = 11539' B = 607'25". c = 8.4177. 

2. a = 365.74, 4. a = 0.062387, 6. a = 31.239, 
6 = 445.84, b = 0.023475, 6 = 49.001, 
c = 545.62. C = 11032'. .1 = 3218'. 

7. Two points A and B are inaccessible from C. If AT? = 1308 ft., 
angle CAB = 537', and angle CBA = 70 15', find the distance from C 
to each of the other two points. 

8. The angles of elevation of a balloon, directly above a straight road, 
from two points of the road on opposite sides of the balloon, are 7815'20" 
and 5947'40". If the two points are 5000 ft. apart, \\ hat is the height of 
the balloon? 

9. A 52-ft. ladder is set against an inclined buttress and reaches 46 ft. 
up its face. If the foot of the ladder is 20 ft. from the foot of the inclined 
face, what is the inclination of the face of the buttress? 

10. A and B are separated by an obstruction, but C is accessible from 
both. If AC = 161.3 ft., ('B = 703.6 ft., and angle C - fKS>'30", 
what is the distance A/*? 

11. A ship sails 23 miles on compass course 15, thence 15 miles on 
compass course 78. How far and in what direction is she from her 
starting point? 

12. The area of a triangle whose angles are 619'32", 3414'46" and 
8435'42" is 680.60. What is the length of the longest side? 

13. The captain of a ship traveling at 14 knots on compass course 66 
sights a lighthouse bearing 39. After 10 inin. the lighthouse bears 
1730'. How long does it take to get to the point nearest the lighthouse, 
and how far away is it at that time?^ 

ft4^ The magnitude h of an inacces- 

sible vertical height DC is desired. A 
base line AB of length d in the hori- 
zontal plane through the base D of the 
object is laid off, and the angles DA C, 
DAB, and DBA are found by measure- 
ment to be a, ft, and ^, respectively 
(see Fig. 18). 
(a) Show that 

h = d sin <p tan a esc (ft -f- <p). 
(b) If d = 132.1 ft., a = 3216 ; , |8 = 2235', <p = 2048 ; , find h. 





71] 



MISCELLANEOUS EXERCISES 



171 



15. From the top of a hill the angles of depression of the top and 
bottom of a flagstaff 25 ft. high at the foot of the hill are observed to be 
45 13' and 47 12', respectively. Find the height of the hill. 

16. The angle of elevation of a balloon ascending uniformly and 
vertically at a height of 1 mile is observed to be 3520'; 20 rniri. later the 
elevation is observed to be 5540'. How fast is the balloon moving? 

17. A flagpole 160.43 ft. high is situated at the top of a hill. At a 
point 600 ft. down the hill the angle between the surface of the hill and a 




FIG. 19. 

line to the top of the flagpole is 8. Find the distance from the 
point to the top of the flagpole and the inclination of the ground to a 
horizontal plane (see Fig. 19). 

18. From a point on a horizontal plane the angle of elevation of the top 
of a mountain peak is 4028'36", and 4163.2 ft. farther away in the same 
vertical plane the angle of elevation is 2850'24''. Find the height of the 
peak above the horizontal plane. 

19. A tower (Fig. 20) stands on a hill inclined 22 with the horizontal. 
At a point A some distance down the hill the angle of elevation of the top 




FIG. 20. 

of the tower is 50 and at B, 200 ft. farther down the hill, the angle is 35. 
Find the height of the tower. 



172 



OBLIQUE TRIANGLE 



[CHAP. VIII 



20. A tower stands at the foot of a hill inclined 18 with the hori- 
zontal. At a point A some distance up the hill the angle of elevation 
of the top of the tower is 28, and at B, 120 ft. farther up the hill, the 
angle is 15. Find the height of the tower. 

21. From a ship two lighthouses bear N. 45 E. After the ship sails at 
11 knots on a course of 130 for 2 hr. ; the lighthouses bear 6 and 356, 
respectively. Find the distance between the lighthouses. 

22. A 50-ft. vertical pole casts a shadow 62 ft. 3 in. in length along the 
ground when the sun's altitude is 4138'. Find the inclination of the 
ground in the line of the shadow. 

23. ^The diagonals of a parallelogram are 376.14 ft. and 427.21 ft., and 
the included angle is 7012 / 38 // . Find the length of the sides. 



24. If R is the radius of a circle circum- 
scribed about the triangle ABC (Fig. 21), 
show that 




2R = 



c 

sin C - 



sin A sin B 
Hint. Angle BAG = angle DOC. 



FIG. 21. 




25. Find the radius of a circle inscribed in a 
triangle whose sides are a, 6, and c (see Fig. 
b 22). 

Hint. The area K of the triangle ABC 
^br + \cr = rs. 



FIG. 22. 

26. Prove that the area K of a triangle is given by the formula 

~ _ dbc 

A- . -_ I 



where R is the radius of the circumscribing circle. 



71] 



MISCELLANEOUS EXERCISES 



173 



27. Show that in any triangle 
(a) a 2 + 6 2 + c 2 
(6) 5JL_ 



2(ab cos C + be cos A + ac cos B) ; 

a 2 + & 2 + c 2 



, cos 7? , cos C 

I 7 l ' ' 



2a6c 



28. An observer whose eye is 37 ft. above the surface of the water 
measures the compass bearing and depression of two buoys as follows: 
A, compass bearing 103, depression 350'; B, compass bearing 165, 
depression 245 / . Find the length AB and the compass bearing of B 
from A. 



29. Find the value of x in Fig. 23. 




Fi. 23. 

30. Two stations, B and C, are situated on a horizontal plane 1200 ft. 
apart. A balloon is directly above a point A in the same horizontal 
plane as B and C y . At B the angle of elevation of the balloon is 6130', 
and the angle at B subtended by AC is 5312', and at C the angle sub- 
tended by AB is 7137'. Find the height of the balloon. 

31. A plane through a vertical flagpole on a small hill contains two 
points A and B lying 130 ft. apart in a horizontal plane, both on the same 
side of the hill. From A the angles of elevation of the top and bottom 
of the flagpole are 13 and 6, respectively, and from B the angle of 
elevation of its top is 10. Find the height of the flagpole. 

32. A, B, C are three objects at known distances apart; namely, 
AB = 1056 yd., AC = 924 yd., BC = 1716 yd. An observer places 
himself at a station P, from which C appears directly in front of A and 
observes the angle CPB to be 1424 ; . Find the distance CP. 

33. The foremast on a freighter sailing west bears N. 35 W. for an 
observer on a submarine 10,000 yd. from the mast. A torpedo fired 
from the submarine in a direction N. 53 W. travels at the rate of 27 
knots and crosses the path of the freighter 235 yd. ahead of its mast. 
Find the speed of the freighter (see Fig. 24 on page 174). (Take 
2000 yd. = 1 nautical mile.) 



174 



OBLIQUE TRIANGLE 



[CHAP. VIII 





FIG. 24. 

34. A vertical plane through the foremast of an anchored freighter 
cuts a hill on the near-by shore in a line AB inclined 37 to the hori- 
zontal. From A the angle of depression of the top T of the mast is 9, 
and from B, 98 ft. downhill from A, the angle of elevation of T is 7. 
If the mast subtends an angle of 14 at B, find its height. 

35. P and Q are two inaccessible objects; a straight line AB, in the 
same plane with P and Q, is measured and found to be 280 yd. long. If 
angle PAB = 95, angle QAB = 4730', angle QBA =- 110, and angle 
PBA - 5220', find the length of PQ. 

36. A and B are two stations 1 mile apart, and B is due east of A. 
When an airplane is due north of A its angles of elevation at A and B 
are 37 and 23, respectively, and when due north of B, its angles of 
elevation at A and B are 12 and 19, respectively. Find its altitude 
at each time of observation and the compass course it is traveling. 

37. On the bank of a river there is a column 200 ft. high supporting a 
statue 30 ft. high. The statue to an observer on the opposite bank sub- 
tends the same angle that subtends a man 6 ft. high standing at the base 
of the column. Find the breadth of the river. 

38. From a certain station the angular elevation of a mountain peak in 
the northeast is observed to be a. A hill 22| south of east whose height 
above the station is known to be h is then ascended, and the mountain 
peak is now seen in the north at an elevation 0. Prove that the height of 
its summit above the first station is h sin a cos ft esc (a 0). 

39. A tower is situated on a horizontal plane at a distance a from the 
base of a hill whose inclination is a. A person on the hill, looking over 
the tower, can just see a pond, the distance of which from the tower is b. 
Show that, if the distance of the observer from the foot of the hill be c, 

* i I.*. J.L x be sin a 

the height of the tower is _,,_, 

a + 6 + c cos a 



71] 



MISCELLANEOUS EXERCISES 



175 



40. The angular elevation of a column as viewed from a station due 
north of it is a, and as viewed from a station due east of the former 
station and at a distance c from it is ft. Prove that the height of the 
column is 

c sin a sin ft 

[sin (a - 0) sin (a + /)]* 

41. An observer found the angle of elevation of the summits of two 
spires which appear in a straight line to be a, and the angles of depression 
of their reflections in still water to be ft and 7. If the height of the 
observer's eye above the level of the water was c, show that the hori- 
zontal distance between the spires is 

2c cos 2 a. sin (ft 7) 
sin (ft a) sin (7 a) ' 

42. A, B, C are three objects so situated that A B = 320 yd., A C = 600 
yd., arid BC = 435 yd. From a station P it is observed that APC 
= 15, and BPC - 30. Find the distances of 

P from A, B, and C if the point A is nearest P 
and the angle APB is the sum of the angles 
APC and BPC. 

Hint. From Fig. 25, PC = 600 sin z/sin 
15 = 435 sin y/siri 30. Solve this equation 
for sin z/sin ?/, apply composition and division, 
and in the result replace sin x sin y by 2 
cos ?(x + y) sin ^(x y) and sin x + sin y by 
2 sin ^(x + y) cos ^(x ?/), and simplify to 
obtain 



tan ?(x y) = 

435 sin 15 - 600 sin 30 l 

435 siri 15M- 600 sin 30 tan * ( * 



(A) 




. 25. 



Compute angle C , replace x + y in (A) by 300 - (15 + 30 + C), and 
solve the result for x y, etc. 

43. Solve a triangle, having given the length of the median to a 
side, and the angles into which this divides the vertical angle. 

44. Three vertical flagstaff s stand on a horizontal plane. At each of 
the points A, B, and C in the horizontal plane, the tops of two staffs 
are seen in the same straight line, and these straight lines make angles 
a, 0, 7 with the horizon. The plane containing the tops makes an angle 
with the horizon. Prove that their heights are BC/[^/cQi^ft cot 2 B 
\/ (cot 2 7 coi^l and two similar expressions. Explain how the 
signs of the roots must be taken. 



176 OBLIQUE TRIANGLES [CHAP. VIII 

45. A certain gun with a shooting range of 1000 yd. per degree of 
elevation is pointed 20 above a horizontal plane. If a direct hit is 
registered on a target at a range of 20,000 yd. when the trunion axis 
is horizontal, find the variation in range and the variation in deflection 
to be expected on the second shot if for it the trunion axis is tilted 
through 5. 

46. Find the answer to the problem resulting when, in Exercise 45, 
the angle of elevation is replaced by 0, the range by R, and the angle of 
trunion tilt by <t>. 



CHAPTER IX 
INVERSE TRIGONOMETRIC FUNCTIONS 

72. Inverse trigonometric functions. To any angle there 
corresponds one and only one value of each trigonometric func- 
tion, but to any value of a trigonometric function there corre- 
spond many angles. Thus sin 30 = ^, but 30, 150, 390, 
and many other angles have a sine whose value is ^. 

The problem of finding the value of a trigonometric function 
of a given angle has already been considered in detail. The 
inverse problem, namely that of expressing the angles when the 
value of a trigonometric function is known, is the problem of this 
chapter. Consider the equation 

y = sin x. (1) 

Evidently x in this equation is an angle whose sine is y. To 
express this we introduce the symbol sin" 1 ,* write 

x = sin" 1 T/, (2) 

and read the symbol sin" 1 y as the angle whose sine is y. Since 
the problem of finding x in equation (1) when y is given is the 
inverse of finding y when x is given, the symbol sin" 1 y is often 
read as the inverse sine of y or the arc sine of y. 

Similarly, the symbol cos"" 1 x means the angle whose cosine is 
x and is read the angle whose cosine is x, the inverse cosine of x, or 
the arc cosine of x. The symbols tan" 1 x, cot" 1 x, sec" 1 x, and 
esc" 1 x are defined and read in an analogous manner. 

Example. Find two positive angles x less than 360 for 
which (a) x = tan" 1 1, (b) x = cos" 1 ( -|-). 

Solution. Since the tangent of a first-quadrant angle or of a 
third-quadrant angle is positive, it appears that x = 46 and 

* In the notation sin" 1 x, 1 is not an algebraic exponent, and sin^ 1 x 
does not denote I/sin x. To avoid confusion, when I/sin x is meant, write 
(sin x)" 1 . 

177 



178 



INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 



x = 225 satisfy x = tan" 1 1. The cosine of a second-quadrant 
angle or of a third-quadrant angle is negative; hence x 120 
and x = 240 satisfy x = cos~ L ( -g-). 

EXERCISES 

For each of the following equations find two positive values of y less 
than 360 satisfying it: 

1. y sin- 1 i. 7. y = cos- 1 ( \/2). 

2. y = sin- 1 i\/3. 8. y = sec- 1 \/2. 

3. y sin- 1 ( Jv^2). 9- // = sec" 1 2. 

4. y = tan- 1 \/3. 10. // = esc" 1 (-2). 

5. y = tan" 1 ( 1). 11. y = csc^irv^- 

6. i/ = cos- 1 (-). 12. // = sin- 1 0.432. 

73. Graphs of the inverse trigonometric functions. Since 

x = sin ?/ and T/ = sin" 1 x 

express the same relation between x and y, we may make a table 
showing corresponding values of x ajid y for plotting y = sin" 1 # 



2;r 



FIG. 1. 




Sf 



Fm. 2. 



by using x = sin y. Since this latter equation is the result of 
interchanging x and y in y = sin x t we can obtain a table of values 



73] 



GRAPHS OF THE INVERSE FUNCTIONS 



179 



for plotting y = sin~ J x by interchanging x and y in the table of 
values used in 46 to plot y = sin x. Hence, interchanging x 
and y in the table of 46, plotting the points represented by the 
pairs of values in this new table, and connecting them by a 
smooth curve, we obtain the graph of y = sin" 1 x (see Fig. 1). 




FIG. 6. 



By a similar procedure tables of values are prepared for plotting 
the other inverse trigonometric functions; their graphs are shown 
in Figs. 2 to 6. 

EXERCISES 

Construct the graphs of the following equations: 

x 

1. y = sin- 1 x- 3. y = tan" 1 2x. 

2. y = cos" 1 ^- 4. y = cot' 1 75' 



180 INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 

6. y = sec- 1 2*. 9 y = 2 tan _ 1 1 

6. y = esc" 1 3s. 

7. 2y = sin- 1 3*. 10 - ^ = 2 cot- 1 &. 

11. y = 9- sec" 1 x. 

8. ?/ = 4 cos- 1 2*. 12 c^-iis. 

74. Representation of the general value of the inverse trigo- 
nometric functions. In 72, we saw that there are generally two 
positive values of x less than 360 satisfying an equation of the 
form 

x = fn~ l (d) (3) 

where fn stands for sin, cos, tan, cot, sec, or esc. If ai and 2 are 
two such values satisfying (3), then 

x = ai + n360 and x = 2 + n360 (4) 

satisfy (3) if n is an integer; for the six trigonometric functions 
of an angle are unaffected when the angle* is changed by an 
integral multiple of 360. When radians are used, the solution 
(4) is written 

x = ai + 2tt7r, and x = a 2 + 2nw. (5) 

Example. Find the general value of sin" 1 ( ^). 
Solution. Expressed in degrees, the two positive angles less 
than 360 each of which has a sine equal to ^-, are 210 and 
330. Hence the general value of sin" 1 ( ^) is 
210 + n360, 330 + n360, 
or, expressed in radians, 

Ibr 



EXERCISES 

1. Find the general value of the angles represented by the following 
symbols : 

(a) sm- 1 ^. _ (g) sin- 1 !. (m) esc" 1 (-2). 

(b) sin^iVS. (h) sin" 1 0.4321. (n) tan" 1 (-1). 

(c) sin-^V^. (i) sin- 1 (-A). (<0 tan' 1 . 

(d) sin- 1 (-K/3). 0') cos- 1 i>/2. (p) cot- 1 1. 

(e) sin^O. (A;) sec" 1 (-\/2). (q) cot- 1 oo. 

(/) sin- 1 (-1). (Z) cos- 1 (-i\/5). W .cot- 1 0.432. 



74] 



REPRESENTATION OF INVERSE FUNCTIONS 



181 



2. For each pair of the following equations, find all values of x that 
satisfy both of them: 



(a) x 
(6) x 



sin- 1 (-), 
tan- 1 A/3, 
(c) x = - 1 



, 

n- 1 i\/2f_ 
c- 1 (- V2 



sin- f_ 

(d) x = sec- 1 (- V2), 

(e) x = esc- 1 2, 
(/) x = cos~ l , 



X = COS" 1 

x = sin- 1 (-1). 
x = tan- 1 (-1). 
x = cot" 1 1. 
x = cot- 1 (-\/3). 



3. Find the general value of the angles represented by the following 
symbols : 



(a) sin- 1 0.36. 

(b) cos- 1 0.60. 

(c) tan~ l 0.90. 
cot- 1 2.1. 
sec- 1 3.42. 



(d) 
(e) 



(/) esc- 1 1.21. 



(g) cos- 1 |. 

(h) sin- 1 i 

(i) tan- 1 . 

0') sec- 1 !, 

(fc) cot- 1 1. 

(/) esc- 1 15. 



4. Show that the general values of tan" 1 a are a + k X 180, where 



a is a particular value. Also show that sin- 1 = k X 180 
cos- 1 = 90 + k X 180 (see Fig. 7). 



and 




FIG. 7. 
5. Using the formulas of Exercise 4, find the general values of when 



(a) 3d = cos- 1 0, 

(b) 50 = sin- 1 0, 



(c) 26 = tan- 1 \/2, 

(d) 30 = tan- 1 \/3. 



In each case write all angles less than 360. 

6. Using the formulas given in Exercise 4, find the general values of 
the angles represented by the following symbols: 



(a) tan" 1 1. 

(b) cot" 1 V3- 



(c) tan- 1 (-1). 

(d) tan- 1 0.342. 



182 



INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 



75. Principal values. Of the many values of an inverse 
trigonometric function, a special one is often called the principal 
value. Many ways of choosing a principal value could be 
devised. The choice dictated by advanced mathematics may be 
obtained by using the following statements. 

Let a represent a positive number throughout this article. The 
principal value of sin" 1 a, cos" 1 a, tan" 1 a, etc., (if it exists) is 
zero or a positive angle no greater than 90. For example, the 
principal value of sin" 1 ^ is 30, that of cos"" 1 1 is zero, and that 
of tan" 1 1 is 45. 

The principal value of sin" 1 ( a) (if it exists) or of tan" 1 ( a) 
is a negative angle no greater numerically than 90. For example, 
the principal value of sin" 1 ( ^) is 30, and that of tan" 1 ( 1) 
is -45. 

The principal value of cos" 1 (a) (if it exists) or of cot"" 1 (a) 
is either 90, 180, or a positive second-quadrant angle. For 
example, the principal value of cos" 1 ( 1/\/2) is 135, that of 
cot" 1 (-1) is 135, and that of cos" 1 (-1) is 180. 

The principal value (if it exists) of sec" 1 (a) or esc" 1 (a) is 
a negative angle lying between 90 and 180. For example, 
the principal value of sec" 1 ( 2) is 120, that of esc" 1 ( \/2) 
is -135, and that of esc" 1 (-1) is -90. 

Figure 8 may help in choosing principal values. In 73, the 
part of each graph drawn with a heavy line is the graph reprc- 



cos 



'V-a) 





\ sin" (-a) 
v taiT l (-a) 



Fia. 8. 

senting the principal value of the associated inverse trigonometric 
function. 



75] 



PRINCIPAL VALUES 



183 



EXERCISES 

1. Find the principal values of the following: 



(a) sin-^v^- 


(g) cot- 


1. 


(6) sin- 1 1\/3. 


(h) cos- 


i- 


(c) sin- 1 0. 


(i) cos- 


i\/2. 


(d) tan- 1 1. 


(j) cos- 


0. 


(e) tan- 1 \/3. 


(k) cos" 


i\/3. 


(/) tan- 1 0. 


(I) csc- 


IV3. 



(m) esc" 1 1. 

(n) cot- 1 \/3 

(o) sec- 1 2. 

(p) cos" 1 1. 

(g) sec- 1 

(r) cot- 1 



2. Find the principal values of the following: 
(a) sin- 1 (-i). (d) tan~ l (-1). 

sin- 1 f ^V (e) tan- 1 (- \/3). 

(/) tan- 1 /'-^ 



(c) sin- 1 ~ 



3. Find the principal values of the following: 
(o)cos->( 7=V (d) cot- 1 (-1). 



(6) cos- 1 ^-^J W cot- 1 (-V3). 

(c) cos- 1 (-i). (/) cot- 1 (-4^) 

4. Find the principal values of the following: 
(a) sec- 1 (-2). (d) sec" 1 ( 1\/3). 



_ 

(6) sec- 1 (-\/2). 
(c) Bec- l (-l). 



(e) esc' 1 (-2). 
(/) csc-M 



(g) esc- 1 (-4 

(*) csc-M-1). 

(i) esc- 1 (tan 135). 



6. Find the principal values of the following: 

(a) sin- 1 (-|). (e) esc" 1 (-\/2). (i) sin- 1 ^ 

(6) tan- 1 !. (/) sec" 1 (-1). 

(c) cot- 1 (-\/3). (0) tan- 1 (sin 270). 

(d) cos" 1 0. (h) cot" 1 1\/3. ^ 

6. Find the principal values of the following: 

(a) sin- 1 (-0.866). (d) sec' 1 (-2.73). 

(6) cos- 1 (-0.414). (e) cot- 1 (-0.472). (h) cos- 1 (-0.913). 

(c) tan- 1 (-1.414). (/) esc" 1 (-6.41). (i) tan' 1 (-13.0). 



0') sec- 1 -\/2- 
(*) cos- 1 (-1). 



(g) sin- 1 (-0.074). 



184 



INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 



7. Using principal values evaluate the following expressions, giving 
your answer in radian measure. 

(a) sin- 1 (i) - sin- 1 (-|). 



(6) sin-'(-l) -sin- 1 ( ^ 
(c) tan- 1 (V5) - tan- 1 ^)- 



O 



(d) cos- 1 (i) - cos- 1 (-i). 

(e) sec- 1 (1) - sec- 1 (-1). 
(/) esc- 1 (-2) - sin- 1 (-). 

8. Verify for principal values the following equations: 

(a) sin- 1 ! +_sin- l l\/3 = -sin- 1 (-!) 

(6) sin- 1 \/^ - 3 sin- 1 i\/3 = -fir. 

(c) sin- 1 (-) + sin- 1 Jv/2 = T^T- 

(d) sin- 1 \/2 - sin" 1 i\/3 = sin- 1 i - |x. 

(e) sin" 1 - + cos" 1 1- = sin" 1 1. 

(/) tan- 1 1 + tan- 1 iV3 = rV - tan- 1 ^/3. 

(g) t&ir 1 oo - sin" 1 !\/2 = tan- 1 \/3 - iV- 

(A) cos- 1 ! + sin- 1 ! = tan- 1 1 + cos- 1 l\/2. 

(i) sin- 1 ^ - cos- 1 (-) = cot- 1 \/3 + sec- 1 (-2). 



76. Relations among the inverse 
functions. Let t be a positive num- 
ber less than 1 and 6 a positive 
acute angle such that sin 6 = t. Fig- 
ure 9 shows a right triangle having 
an angle equal to 0, the hypotenuse 
flquai to 1, the leg opposite 6 equal 
to t, and the leg adjacent to 6 equal to 

l t 2 . From the figure we read 





or 



e = esc- 1 T- 



77] EXAMPLESINVERSE FUNCTIONS 185 

Since all these values of are equal, we have for principal values 

I 

sin" 1 1 = cos" 1 \/l t 2 = tan" 1 7 = csc -1 l/t 



= cot- 1 



Hence, for principal values, we have the following relations: 



sin" l u = csc~ l -j 



cos" 1 u sec" 1 > 
u 

provided u is a positive number loss than 1, and 

tan- 1 u = cot- 1 -i 
u 

when u is any positive number. 

77. Examples involving inverse trigonometric functions. The 

solutions of many trigonometric equations are effected by employ- 
ing the relations existing among y 
the inverse trigonometric functions. 
When solving an equation involving 
inverse functions, the student will 
find it advantageous to draw a right 
triangle for each of the angles involved 
in the original equation, and designate 
the lengths of the sides appropriately. 
From these triangles the value of any desired trigonometric func- 
tion is taken directly. The following examples will illustrate the 
method. 

Example 1. Find the value of cos (sin~ l -if-) using the principal 
value of sin* 1 --. 

Solution. Let a represent the principal value of sin" 1 -jj-. 
The right triangle exhibiting a is shown in Fig. 10 with the sides 
appropriately numbered. From this figure we read directly 



4 
FIG. 10. 



cos (sin" 1 



= cos a = - 



186 



INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. L\ 



Example 2. Using principal values for the inverse functions 
involved, find 



cos [cos- 1 ( -J-) + sin- 1 ( -J-)]. 



(a) 



Solution. Let a represent the principal value of cos" 1 ( -g-) 
and ft the principal value of sin" 1 ( -J-). Substitution of these 
values in (a) gives cos (a + ft). Expanding this, we obtain 

cos a cos ft sin a sin ft. (6) 

Consider the two right triangles in Fig. 11, one exhibiting angle 
a, the other angle ft. In accordance with the definitions of 

Y Y 




VI5 



(a) 




FIG. 11. 



principal values we must take a in the second quadrant and ft 
in the fourth quadrant. 

Reading the values of cos a, cos ft, etc., direct from the triangles 
and substituting them in (6), we obtain 



/ 1\ (VI8\ _ (2V2\ / 1\ _ 
\ 3>\ 4 ) V"8~A V 



Example 3. Show that 
tan- 1 (~\ + sin- 1 (~7 



= i cor 1 (-0.6) - 90 (a) 



provided principle values for the inverse functions are used. 

Solution. Let A = tan" 1 (-f), B = sin- 1 (-1/VT7), C = 
cos" 1 ( 0.6). From these and the conventions of 75, it appears 
that angles A, B, and C are correctly represented in Fig. 12. 
Inspection shows that the two members of equation (a) are 



77] 



INVERSE FUNCTIONS 



187 



negative acute angles. Hence they are equal if a trigonometric 
function of one member is equal to the same trigonometric 




FIG. 12. 

function of the other. Equation (a) may he written 

A + B = <7 - 90. 
The cosine of the left-hand member of (b) is 

cos (A + B) = cos A cos B sin A sin B, 
and the cosine of the right-hand member of (I) is 
C - 90) = sin C 



cos 



- osC). 



(c) 



(d) 



Replacing the functions in (c) and (d) by their values read from 
Fig. 12, we have 



cos (A + B) = (\ (*\ - (- ~L\ (-^L 
VV85/VV17/ VV85AV17 




cos 



Since these values are equal, equation (a) is true. 

EXERCISES 

Using principal values for the inverse functions involved, evaluate the 
following expressions : 

1. sin (sin- 1 1). 6. sin [seer 1 (-f)l. 11. tan [cor 1 (1)]. 

2. cos (cos- 1 f). 7. cos [esc- 1 (-)]. 12. sec [cor 1 (5.4)]. 

3. sin (cos- 1 A). 8. cos [cor 1 (-f)]. 13. cos (2 tan"" 1 1). 

4. cos (sin" 1 1). 9. cos [tan- 1 (-)]. 14. tan (cos- 1 f). 
6. esc [tan- 1 (-\/7)]. 10. sec (cor 1 2). 15. sin (cor 1 1). 



188 INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 

16. Evaluate the following expressions, using principal values: 

(a) tan [tan" 1 + tan" 1 (~f)]. 

(b) sec (cos" 1 ^ sin" 1 ). 

(c) esc [sin" 1 (1/V2) + tan" 1 1]. 

(d) sin [sec- 1 (-2) - sin- 1 (-)]. 

Using principal values for the inverse functions involved, verify the 
following equations : 

17. sin- 1 1 - tan- 1 1 = |- 

18. tan" 1 ^r + tan" 1 f = tan" 1 i + tan" 1 . 
Hint. Take the tangent of both members. 

19. tan" 1 i + sin" 1 iVVlO = ITT. 

20. sin" 1 | + sin" 1 fr + esc" 1 ff = esc" 1 1. 

Hint. Transpose esc" 1 fir to the right member and take the cosine 
of both members. 

21. cos" 1 n + tan" 1 1 = cot" 1 M- 

22. tan" 1 jV + tan" 1 &% = sec" 1 i\/5 

23. cot" 1 7 + tan" 1 i + cot" 1 18 = cof" 1 3. 

24. tan" 1 |f - cot" 1 4 = 2 tan" 1 i. 
26. tan" 1 f + tan" 1 i = i sec" 1 f . 

26. sin- 1 -~= + sec" 1 -j^ + esc- 1 2 = y^. 

27. cos (2 sec" 1 }\/93) = sin (4 sin" 1 iVVlO). 

Cftf 2 tan" 1 i + tan" 1 y = iir. (Clausen's formula for finding the 
value of ?r.) 

29. 4 tan" 1 tan" 1 ij-g-ff = i?r. (Machines formula for finding the 
value of TT.) 

30. tan" 1 v?v = tan" 1 Tnr tan" 1 A. 

31. tan- 1 f + tan" 1 i = ITT. 

32. cor 1 3 + esc" 1 V5 = i?r. 

33. tan- 1 a; + tan- 1 y = tan- 1 ^ + ^ 

i a/y 

34. 3 sin- 1 x = sin- 1 (3z - 4z 3 ), -^ ^ a; ^ i 

36. sin (2 sin- 1 x) - 2x\/T ::: "x i , -1 ^ ^ 1. 

Find the value of the following expressions in terms of a and 6; 
assume a and 6 positive, and use principal values for the inverse functions 
involved. 



sec" 1 a cos" 1 r 



78] TRIGONOMETRIC EQUATIONS 189 

36. sin (2 cos" 1 a + ? cos" 1 6). 

37. cos ( 

38. tan ( csc" 1 h esc" 1 TI- 

\ a bj 

39. sin j 2 cos- 1 ftan (| - 2 tan- 1 <*)][ 

40. Solve Exercises 36 to 39, assuming that both a and 6 are negative. 

78. Trigonometric equations. An equation which involves 
one or more trigonometric functions of a variable angle is a 
trigonometric equation. A trigonometric identity is a trigono- 
metric equation which holds true for all values of the variable 
for which the members of the equation are defined. On the other 
hand, a trigonometric equation which is satisfied by only particu- 
lar values of the variable is a trigonometric equation of condi- 
tion. The problem connected with an identity concerns the 
proof that it is invariably true, whereas the problem associated 
with an equation of condition is to discover for what values it is 
true. By a solution of a trigonometric equation we mean general 
expressions defining all values of the variable which will satisfy 
the given equation. This will mean in many problems that a 
number n representing any integer must be used. 

There are a number of methods for solving trigonometric 
equations. It is often possible to express all trigonometric func- 
tions involved in terms of a single function, sol ( ve the resulting 
equations for this function, and then write the angles associated 
with the values of the function. Another method consists in 
transferring all terms of the given equation to the left-hand 
member, factoring the resulting left-hand member, equating the 
factors to zero, and solving each equation thus obtained. The 
following examples will illustrate these methods of procedure. 

Example 1. Solve 2 cos 2 x + sin x 1 = 0. 
Solution. Replacing cos 2 x by 1 sin 2 x and simplifying 
slightly, we obtain 

2(sin xY - (sin x) 1 - 1 = 0. 
Evidently this is a quadratic equation with sin x appearing as the 



190 .INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 

unknown. Solving it by formula,* we obtain 



Hence x = sin~ l 1 and x = sin" 1 ( -J-). Replacing those inverse 
functions by their general values, we get 

x = 90 + /i360, x = 210 + n360, x = 330 + n360 
or, in radians 

X = s + 2nir, # = ^r + 2/iir, # = -~ h 2n?r. 

Example 2. Solve sin 40 + cos 26 = 0. 

Solution. Replacing sin 40 by 2 sin 20 cos 20 in the given 
equation and factoring, we obtain 

cos 20 (2 sin 20 + 1) = 0. 
Equating the factors to zero, we get 

cos 20 = 0, 2 sin 20 + 1 = 0. 
From cos 20 = we derive 

20 = 90 + n360, and 20 = 270 + ^360. (a) 
or 

= 46 + n!80 and = 136 + n!80. 

From 2 sin 20 + 1 = 0, or sin 20 = ^, we derive 

20 = 210 + n360 and 20 = 330 + n360, 



or, 



= 106 + n!80 and = 165 + n!80. 



EXERCISES 

1. Find the values of x between and 360 for which 

(a) sin 2 x = i. (d) sec 2 x - 4 = 0. 

(6) esc 2 x = 2. (e) tan 2x = 1. 

(c) tan 2 x - 3 = 0. (/) 2 sin 3x = 1. 

* The solution of ay 2 + by + c is y = := - 



78] TRIGONOMETRIC EQUATIONS 191 

2. Find the values of the unknown between and 360 for which 

(a) 2 sin 2 x + 3 cos x = 0. (e) 4 sec 2 y - 7 tan 2 y = 3. 

(b) cos* a - sin 2 a = . (/) tan 5 + cot B = 2. 

(c) 2\/3 cos 2 a = sin a. (0) sin x + cos x = 0. 

(d) sin 2 1/ 2 cos y + } = 0. 

3. Find, in radians, all angles between and 2ir that satisfy the follow- 
ing equations: 

(a) (tan .r + l)(\/3 cot x - 1) --= 0. 

(b) (2 cos x + D(sin x - 1) = 0. 

(c) (4 cos 2 - 3) (esc + 2) - 0. 

(d) 2 cot sin + cot 6 = 0. 

4. For each of the following equations, find all values of the unknown 
that satisfy it: 

(a) 2 sin 2 x + cos x - 1 = 0. (A;) tan 2 x + cot 2 x - 2 = 0. 

(6) 2 cos 2 + 5 sin 0-4 = 0. (0 tan x + 3 cot x = 4. 

(c) cos 2 JT + 2 sin .r + 2 = 0. (m) 2 tan 2 x + 3 sec a: = 0. 

(d) 2 cos 2 2a + sin 2a - 1 = 0. (n) cos + 6 sin = 2. 

(e) 2 sec 2 tan = 5. (o) sin x + cos x = 1. 
(/) 2 esc 2 < - 5 cot + 1 = 0. (p) esc a; cot x = 2^3- 
(flf) 4 sec 2 2A = 8 + 15 tan 2A. (q) sin x cos x + i = 0. 
(/i) cos 2 x(4 cos 2 x 1) = 0. (r) cos 2x + cos x = 1. 
(i) 4 cos 2x + 3 cos x = 1. (s) tan 20 tan 0=1. 

(j) cot 2 - 3 esc + 3 = 0. 

6. Solve for the unknown: 

(a) 2 sin = tan 0. (/) sin 20 = V3 J.'.n 0. 

(6) sin 2x cos x = 0. (0) sin 2 4a = sin 2 2a. 

(c) 4 sin 4 = 3 sin 2 0. (h) 2 sin 40 + sin 20 = 0. 

(d) sin 2a + cos a = 0. (i) cos 4a = cos 2a. 

(e) sin 4x = cos 2x. 

6. Find the abscissas of the points where each of the following curves 
crosses the x-axis: 

(a) y = 2 sin x sin 2x. (c) y = cos 2x cos 2 x. 

(b) y = cos 2x cos x. (d) y = tan (x + 45) 1 + sin 2x. 

7. Plot each of the following pairs of curves on the same set of axes 
and find their points'of intersection for values of x between and 360. 

(a) y = sin 2x, y = sin x. 

(b) y = cos 2x, y = cos x. 



192 INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 

(c) y = sec x, y = 2 cos x. 

(d) y = tan x, y = 3 cot x. 

(e) y = 2 sin a;, y = tan . 

(f) y = tan 2 x, y = 2 cot 2 x. 

79. Special types of trigonometric equation. The solution of 
certain types of trigonometric equation may often be obtained 
by transforming the equation or by some other device. The 
following examples will illustrate two methods. 

Example 1. Solve cos 6z = cos 4x for x. 
Solution. Write the given equation in the form 

cos 6x cos 4x = 0, 
and apply the conversion formula 

cos A - cos B = -2 sin %(A + B) sin %(A - E) 
to the left-hand member and get 

-2 sin (6z + 4x) sin ^(6z - 4z) = 



or 

2 sin 5x sin x 0. 

Equate the factors sin 5x and sin x to zero and obtain 

sin 5x = 0, sin x = 0. (a) 

From the first of equations (a) we get 

5z = + n360, and 5x = 180 + n360 
or 

x = n72 and x = 36 + n72. 

From the second of equations (a) we get 

x = + n360 and x = 180 + n360. 

Example 2. Solve sin 9x = cos 4x for x. 
Solution. Write the given equation in the form 

sin 9x - sin (90 - 4z) = 0, 

t 
and apply the conversion formula 

sin A - sin B = 2 cos $(A + B) sin -J(A - B) 



79] SPECIAL TYPES OF TRIGONOMETRIC EQUATION 193 
to the left-hand member and obtain 

2 cos (f z + 45) sin (x - 45) = 0. 
Set the factors equal to zero and get 

cos (fc + 45) = 0, sin (*f* - 45) = 0. (a) 

From the first of equations (a) we get 
f z + 45 = 90 + n360, and f z + 45 = 270 + n360, 

or 

x = 18 + n!44, and x = 90 + n!44. 

From the second of equations (a) we get 

lx - 45 = + n360 and l$-x - 45 = 180 + n360 

or 



x = 



90 + n720 
13 



460 + n720 
13 



In accordance with Exercise 4, 74, the complete answer could 
be written in the form 

90 + 360n 
13 



x = 18 + n72, x = 



Example 3. Solve 7 sin 3x 1 1 cos 3x 
12 for x. 

Solution. To solve this equation first 
transform the left-hand member into the sine 
of the difference of two angles. To do this 
let a = tan" 1 ty, and construct Fig. 13. Divide the given 
equation through by \/170 to obtain 




VT70 



o 
sin 3x 



VT70 



cos 3x = 



12 



-V/170 



(a) 



In (a) replace 7/Vl70 by cos a and 11/VTTO by sin a, their 
values from Fig. 13, to get 



sin 3x cos a cos 3x sin a 



12 



(6) 



194 INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 

or 

sin (3x a) = ; 

A/170 

Use the slide rule or natural function table to obtain 

a = tan- 1 A^ - 5732', sin- 1 = = 6659', and 

113!'. (c) 
Use these angles to get 

3* - 5732' = 6659' + n360, (d) 

3z - 5732' = 113!' + w380. (0 

Solve (d) and (e) for a: to obtain 

x = 4130 / + nl20, a- = 6651 / + nl20. 

EXERCISES 

1. Solve for the unknown : 

(a) sin 30 - sin 90 = 0. 
(6) cos 60 = cos 20. 

(c) sin MX = cos 7z. 

(d) sec 9x = sec 5x. 

(e) tan 4# = cot 6x. 
(/) sec 8x = esc lOz. 

(^) 4 sin x + 3 cos .r = 1. 

(h) 3 sin - 4 cos = 3 

(i) 12 cos a + .5 sin a = 6.5. 

(j) 5 cos </> 12 sin <j> = 3|. 

(fc) cos 2x 2 sin 2x = 2. 

(/) 12 sin 30 - 5 cos 30 = 5. 

(m) sin 4x sin 2x cos 3x = 0. 

(n) cos 50 + cos 30 + cos = 0. 

(o) sin 40 = sin 90 - sin 0. 

(p) 2 sin 30 cos 0-2 sin cos 30 + 1 = 0. 

(q) tan 40 = tan 100. 

(r) 2 sin A cos A 2 cos ^4 + sin .1 1 = 0. 

(s) 3 sin + cos = 2x, sin + 2 cos = x. 

2. Solve the equations 

r cos < cos = 2, 

r cos <t> sin = 3, 

r sin </> = 5. 



EQUATIONS INVOLVING INVERSE FUNCTIONS 195 

Hint. Divide the first equation by the second, member by member. 

3. Solve the equation 

sin (a + x) = m sin x, 

for tan (3 + i). 

4. Solve the equations 

m sin (0 + x) = a, 
m sin (</> + #) = b, 

for m and x, the other four quantities, 0, <, a, 6, being known. 

Hint. Expand sin (^ + #), sin (6 + x) and solve for sin x and cos x. 

5. Solve m cos (0 + z) = a, and m sin (<t> -\- x) = 6, for m sin x and 
w cos x. 

6. Solve m cos (0 + x) = a, and m cos (< x) = 6, for wi sin x and 
m cos x. 

7. Solve the equations 

x cos a + ?/ sin a = m, 
sin a ?/ cos a = n, 

for x and y. 

i 

80. Equations involving inverse functions. The following 
example will furnish an illustration of the method of solving an 
equation involving inverse trigonometric functions. In solving 
problems of this type, we shall understand that principal values 
only are to be considered. 

Example. Solve the equation 



11-10 * i - 5z 2 + 1 , v 

cos" 1 x + sin" 1 2x = tan" 1 - , (a) 

x \/5 ~ 8x 2 
Solution. If we let 

cos" 1 x = a. sin" 1 2x = ft. tan" 1 - . = 7 

x V 5 8x 2 

and if we substitute these values in (a), we have 

a + ft = -7. 

Taking the cosine of both members of this equation, we obtain 
cos (a + ft) = cos 7, 



196 



INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 



or 



cos a cos |8 sin a sin ft = cos 7. 



(6) 



The three triangles exhibiting a, 0, and 7 are shown in Fig. 14. 
Reading direct from the triangles the values of the functions 




x 
(a) 




2x 




Yl-4x 2 
(6) 

FIG. 14. 



(c) 



involved in (b), and substituting these values in (b), we obtain 

la:) = a?\/5 8x 2 . 



=~& - VI - 
Solving this equation, we get 
z = 0, a: = 



and 



= 1. 



Substituting these values of x in the original equation, we find 
that only x = % satisfies it for principal values. Hence the 
solution is 



EXERCISES 

1. Verify that x = ^ does not satisfy (a) of the foregoing example if 
principal values only are considered. 

2. Solve the following equations for the unknown, using principal 
values only: 

(a) sin- 1 y + sin- 1 2y = * 

(6) tan- 1 2x + tan' 1 3x = -j-' 

(c) tan (sin- 1 \/l - s 2 ) - sin (tan- 1 2) = 0. 

(d) tan" 1 y = sin" 1 a + cos- 1 6, 1 > 6 > and, numerically, b > a. 



81] MISCELLANEOUS EXERCISES 197 

(e) 2 tan- 1 y = ~ - cot- 1 3y. 

(/) 2 tan" 1 + cos" 1 f = sin" 1 

x 

(g) tan" 1 x + tan" 1 (1 x) = 2 tan- 1 -\/x(l x). 
(ft) sin- 1 - + sin- 1 - 

/x i m , i n * 

(i) sm- 1 - + sin- 1 - = 2" 

0') sin" 1 x = 2 cos" 1 x. 
(k) sin- 1 x = 2 tan" 1 x. 
(I) tan- 1 x = 2 sin- 1 x. 
(m) cor 1 x - cor 1 (x + 2) = 15. 

(a sin- 1 x + 6 cos- 1 t/ = a) 
(n) \a cos- 1 x - b sin- 1 y = 0/' 

81. MISCELLANEOUS EXERCISES 

1. Find the values of the following: 

(a) sin (tan" 1 rs)- 

(b) sin (tan" 1 ? + tan" 1 i). 

(c) tan (2 tan- 1 a). 

(d) cot (2 arc sin f). 

(e) cos (2 arc cos a). 
(/) cos (2 arc tan a). 

(g) arc tan j=* 
V3 

(ft) cor 1 (1). 

2. Prove the following using principal values: 

(a) tair 1 1 + tan" 1 2 + tan" 1 3 = TT. 
(6) arc cos y + arc tan | = arc tan TT 

(c) 2 tan" 1 = tan" 1 ^. 

(d) sin" 1 f + sin" 1 TT = sin" 1 ff. 

(e) arc cos f + arc cos T| = arc cos ff. 
(/) arc tan y + arc tan riar = arc tan f . 

Solve the following equations: 

3. (a) sin x = 3 cos x. 
(6) 2 cos x = cos 2x. 
(c) tan x = tan 2x. 



198 INVERSE TRIGONOMETRIC FUNCTIONS [CHAP. IX 

4. @ 3 cos 2 x + 5 sin x 1 =0. 
(6) 3 sin x tan x 5 sec x + 7 = 0. 
fy tan x + sec 2 x 3 = 0. 

(d) sin x + cos 2x = 4 sin 2 a; 1. 

(e) sin (2x - 180) = cos x. 
(/) cos 2 x + 2 sin x = 0. 
(g) sec 2 a? 4 tan x = 0. 

(A) sin 2 2x sin 2x - 2 = 0. 

(i) tan 2 I - tan | - 2 = 0. 

0') sin x sin s = 1 cos x. 

(k) esc y + cot y = \/3- 
(I) 6 sec 2 a + cot 2 a = 11. 
6. (a) cot 5x = cot 7x. 

(b) sec 3x = esc 5x. 

(c) sin 3x sin x = sin 5x. 

6. cos 5x + cos 6x = sin 5x + sin 6x. 

7. (a) 4 sin x + 3 cos x = 3. 
(6) 5 sin x = 4 cos x + 4. 

8. (a) sin (60 - x) - sin (60 + x) = ^- 

(6) sin (30 + x) - cos (60 + x) = - ~- 

(c) tan (45 - x) + cot (45 - x) = 4. 

(d) sec (x + 120) + sec (x - 120) = 2. 

(e) esc 2 x(l + sin x cot x) = 2. 

9. (a) sin x + sin 2x + sin 3x = 0. 

(b) tan x + tan 2x + tan 3x = 0. 

(c) sin 4x cos 3x = sin 2x. 

x 2 y 2 
10. (a) If x = a cos <p, y b sin ^, prove that 2 + r^ = 1. 

(I 

Hint. Solve for sin <p and cos <p and then use siri 2 <p + cos 2 <p = 1 . 

xj.2 y2 

(b) If x = a sec ^, y = a tan ^, prove that -^ T 2 = 1. 

(c) From x = a cos 3 <p, y = a sin 3 ^>, deduce x* + 3/3 = at. 

(rf) If x = a + ft cos ^>, y = c + d sin ^, find a relation between 
x and i/. 

(e) From x = a tan 3 ^>, i/ = ft sec 3 <p deduce a relation between 
xand v. 

|jn If a sin + ft cos B = />,, a cos ft sin = A;, prove that 



81] MISCELLANEOUS EXERCISES 199 

11. Solve the following equations : 

(?o) tan- 1 x + tan- 1 (1 - x) = tan- 1 (|). 

2w 
(6) arc tan x + 2 arc cot x = -*- 

o 

/ \ A i x 1 A ,3 + 1 W 

(c) tan- 1 ^qp - 2 + tan- 1 ^-^ = y 

/^ i * 2 - * j_ * i 2x 2?r 
(^ cos- 1 ^-^ + tan- 1 - 2 - x = T - 

(e) arc tan r + arc tan = arc tan ( 7). 

x i x 

(/) tan- 1 (x + 1) + tan- 1 (3 - 1) = tan" 1 / T - 
(g) sin" 1 x + sin- 1 2x = 5- 

o 

/L\ ' 5 1 . 12 T 

(h) arc sin h arc sin = ~- 

X X t 

12. Plot each of the following pairs of curves on the same set of axes, 
and find their points of intersection between and 360. 

(a) y = sin x, y tan x. 

(b) y = 2 sin x, y = tan 2x. 

(c) y = tan x, y = 4 3 cot x. 

(d) y = cos 2x, y = -(1 + cos x). 



CHAPTER X 
COMPLEX NUMBERS 

82. Pure imaginary numbers. In algebra it was found neces- 
sary to extend the number system to include imaginary numbers. 
A pure imaginary number is the indicated square root of a nega- 
tive number. Thus v5 is a pure imaginary number. 

It is customary to reduce a pure imaginaiy number to the form 
6A/ 1 where b is a real number,- to substitute the letter i for 
V 1, and then to treat i as a literal algebraic quantity that 
obeys all the laws of algebra in addition to the law i 2 = 1. 
It follows that a power of i is equal to one of the following: 
i, 1, i, 1. Thus 



i 5 = i*i = 



EXERCISES 

1. Express each radical in terms of i and simplify, noting that 




if P is real and positive. 
(a) \^~-36. 



(6) V-T. (g) -16o; 2 . (A) Vac, 4ac 

(c) \^49. (/) V- 

2. Write the two square roots of each of the following quantities: 
(a) -16. (b) -9x 2 . (c) -13. (d) -7a 4 a; 2 

3. Simplify 

(a) i". (c) t (e) ti. (j;) i>3. 

() t* w . (d) . (/) H'". (A) ~- 9 - 

200 



84] OPERATIONS INVOLVING COMPLEX NUMBERS 201 

83. Complex numbers. A complex number is one having the 
form a + bi where a and b represent real numbers and i = \/ i; 
bi is termed the imaginary part. Any real number may be 
considered as a complex number in which the coefficient b of i 
is zero. 

Two complex numbers are said to be equal if their real parts are 
equal and their imaginary parts are equal. Thus a + bi = c + di 
if a = c and b = d. Conversely, if a + bi = c + di, then 
a = c and b d. It therefore follows in particular that, if 
a + bi = 0, then a = and 6 = 0. 

In what follows we shall find it convenient to use the term 
conjugate complex number. Two complex numbers that differ 
only in the signs of their pure imaginary parts are called conjugate 
complex numbers. Thus (2 + 30 and (2 3z) are conjugate. 

84. Operations involving complex numbers. Since i obeys all 
tlio laws of algebra and since a and 6 are real numbers, we may 
operate with the complex number a + bi in the usual way. In 
adding (and subtracting) complex numbers, it is necessary to add 
(or subtract) the real parts and the imaginary parts separately. Thus 

(4 + 60 + (5 - 70 = [4 + 5 + (6 - 7)i] = 9 - i, 
(7 - 20 - (9 + 40 = [7 - 9 - (2 + 4)<] = -2 - 6i. 

In performing a multiplication one should replace i 2 by 1 
whenever i' 2 occurs. Thus 

(6 - 50(9 + 20 = 54 + 12z - 45t - 10i 2 = 64 - 33i. 

The quotient of two complex numbers can be obtained in the form 
a + bi by multiplying both numerator and denominator by the 
conjugate of the denominator. Thus 

4 - 7i = (4 - 7Q(6 - Q = 24 - 4i - 42i + 7i* 
6 + i (6 + 0(6 - " 36 - i 2 

_ (24 - 7) - 46i 17 _ 46. 
37 37 37 1 ' 

EXERCISES 

1. Find real values for x and y if 

(a) x + yi = 2 - 3i. (c) (3* - 2) - (4 - y)i - 0. 

(6) 3z - 2yi = 5 + 7t. (d) 2x - 4yi = 6 - 2si'. 

(e) 7x + Qy + 2xi - 3yi + 9 = x + yi - y + 3 - 2t. 



202 



COMPLEX NUMBERS 



[CHAP. X 

2. Write the conjugate of each of the following complex numbers: 
(a) 7 + 2i. (b) x - yi. (c) 3i. (d) 14. 

3. Perform the indicated operations. 



(a) (2 - 50 + (3 + 4i). 

(6) (7 - 50 - (11 - 130- 

(c) (2 + 3<) + (4 - Cn). 

(d) (2 + 30 + (1+ 0. 



(e) (3 - 50 + (3 + 50. 
(/) (6 + 00 - (3 - 70. 
(g) (4 + 20 + (-2 -40. 
(h) (3 + 40 - (3 - 40. 



4. Show that the sum of two conjugate complex numbers is a real 
number arid that the difference is a pure imaginary number. 
6. Perform the indicated operations. 



(a) (3 + 50(6 - 2i). 

(b) (4i - ). 

(c) (2 - 4i)(-3 + 20. 



(d) (7-40(7 + 40. 

(e) i(2 - 50. 

(/) (7 - f)(l + 0(1 - 40. 



6. Show that the product of two conjugate complex numbers is a real 
number. 

7. Reduce the following quotients to the form a + hi. 



4 7i 



(b) 3 + *'. 

(b > 2+ i 
2_ 

V ' /) _ 'W 



I 



(3 



(3 - 20(1 + 



_ . 

0(2-3i) 
_7iK8 +JK) 
(5 -70(4"+ 60' 

(4 -- 5 -^- 

1(6 - 80 



85. Geometrical representation of complex numbers. In 
y 19 it was pointed out that all Teal 

numbers may be represented by 



B(-3,7) 



tt 



ff(0,2> 



-2^5)-^J?^~6) 



ssa 



[F(7, 



0) 



y 



>x 



points on a straight line. Since 
complex numbers depend on two 
real numbers, it, is necessary to use 
two dimensions in order to represent 
a complex number graphically. 
Accordingly, using the system of 
rectangular coordinates explained in 
Chap. Ill, we may represent the 
complex number x + yi by a point P 

whose coordinates are x and y. The x-axis is called the axis of 
reals, and the y-axis the axis of imaginaries. Evidently a real 



Fia. 1. 



POLAR FORM OF A COMPLEX NUMBER 203 

number is plotted on the axis of reals and a pure imaginary 
number is plotted on the axis of imaginaries. 

For example in Fig. 1, point P (x, y) represents the complex 
number x + yi\ point A (5, 7) represents 5 + li\ B( 3, 7) repre- 
sents -3 + 7t; C(-2, -5) represents -2 - 5t; D(2, -6) 
represents 2 6i; JE(0, 2) represents the pure imaginary number 
2i and, F(7, 0) represents the real number 7. 

EXERCISES 

1. Represent graphically the following complex numbers: 

(a) 3 - 2i. (b) -4 + i. (c) Ci. (rf) 0. (c) 1 - >/-2. 

2. Plot the conjugates of the numbers in Exercise 1. 

3. Find the sum of the numbers in Exercise 1 and plot the result. 

86. Polar form of a complex number. Complex numbers can 
be represented in another form involving 
trigonometric functions. In Fig. 2 let 



P(XJ y) represent the complex number 
x + yi. Connect P with the origin of 
coordinates; denote J)y r the length of the 
connecting line OP and by 6 the angle 
that OP majces with the axis of reals. 
Then P(x, y) is determined by r and 8. 
From the figure we have 



>y 






x = r cos 8, y = r sin 8. 

Replacing x and y in x + yi by these values, we obtain 
x + yi = r(cos + i sin 0). 

The form r(cos 8 + i sin 8) is called the polar form of a complex 
number. The angle 8 is called the amplitude and the length r 
the modulus. Here r is positive and 8 is any angle that is gener- 
ated by the positive half of the x-axis when it is turned about the 
origin until its terminal position passes through P(x, y). From 
this it appears that if a is one amplitude of a complex number, 
the other permissible amplitudes are (a + 2irn), where n is any 
integer. 



204 



COMPLEX NUMBERS 



[CHAP. X 



In finding the values of r and it is well to solve* the right 
triangle of which the lengths x and y are the legs (see Fig. 2). 

For convenience some writers use the notation cis 6 as an 
abbreviation for cos 6 + i sin 6. We shall use this notation 
occasionally. 

Example. Write the complex number 4 + 3i in the polar 
form. 

Solution. We first plot 4 + 3i and form the right triangle 

shown in Fig. 3. Solving this tri- 
angle in the usual way (128, 127) 
we find that r = S and a = 3652'. 
The amplitude is found from the 
* figure to be = 180 - a = 1438'. 

_j >x Hence, using the notation cis 6 for 

cos 6 + i sin 6, we have 



-4 



FIG. 3. 



-4 + 3i = 6 cis 1438'. 



If the slide rule is not used for solving the triangle, we may 
write 



r = V( - 3) 2 + 4 2 = 5 



and 



= tnn- l (-f) = 1438'. 



Evidently the amplitude may be taken as (1438' + rc360), 
where n is any integer. 




EXERCISES 



1. Write both forms of the com- 
plex number represented by point P 
of Fig. 4. 



Fio. 4. 



2. Write the polar form of the complex number represented by the 
point P(l, 1). 

* For the method of solving a right triangle by means, of the slide rule, 
see 127, 128. 



87] MULTIPLICATION OF COMPLEX NUMBERS 205 

3. Plot the following complex numbers and write them in the form 
x + yi: 

(a) 2(cos 30 + i sin 30). (e) 11 cis 210. 

(b) 3(cos 60 + i sin 60). (/) 7 cis 270. 

(c) 2(cos ITT + i sin |TT). (g) 6 cis 300. 

(d) 4(co's 180 + i sin 180). (h) 6 cis 60. 

4. Write the following complex numbers in the polar form: 

(a) 1 - i. (/) 5. (k) 7 - bi. 

(b) -2 - 3i. (g) 7i. (0 3.2 - 5.4t. 

(c) -2 + 3i. (A) 0.7 + 1.1 1. (m) -6.1 + 4.2i. 

(d) 4 + Oi. (i) 3/(2i). (n) -3.3 - 6.61. 

(e) + 4i. 0") -i. (o) 7.1 - 4.4*. 

87. Multiplication of complex numbers in polar form. Multi- 
plying the two complex numbers ri(cos a + i sin a) and r 2 (cos + 
i sin ]8) in the usual way, we obtain 

ri(cos a + i sin a) r 2 (cos + z sin 0) -- 

= rir 2 (cos a cos ft + i sin a cos ft + i cos a sin sin a sin 0) 

= r*ir 2 [(cos a cos ft sin a sin 0) + z'(sin a cos )3 + cos a sin 0)]. 

This can be reduced, by using formulas (1) 52, to 

rir 2 [cos (a + 0) + i sin (a + fl)]. 
Using the notation cis 9 for cos + i sin 0, we may write 

(ri cis a)(r 2 cis g) = rif 2 cis (a + g). (1) 

Or, stated in words, 

The modulus of the product of two complex numbers is the product 
of their moduli, and the amplitude of the product is the sum of their 
amplitudes. 

By using this italicized statement with the first two of three 
complex numbers we get 

[ri(cos ai + i sin i)r 2 (cos c* 2 + i sin 2 )]r 3 (cos 3 + i sin a 8 ) 
= rir 2 [cos (<*i + 2 ) + i sin (on + a 2 )]r3(cos 3 + i sin a), 

and this last line is equal to 

rir 2 r s [cos (ai + 2 + a 3 ) + i sin (on + 2 + as)]. 



206 COMPLEX NUMBERS [CHAP. X 

Continuing this process repeatedly for the product of n complex 
numbers, we should finally obtain 



r n ) cos [(i + 2 + * a n ) + i sin (i + 2 + )] 
Using the notation cis for cos 8 -\- i sin 0, we may write 

cis a)(r 2 cis a 2 ) (r cis n ) 

r n ) cis (a t + a 2 + + o n ) (2) 



or, stated in words: 

The modulus of the product of n complex numbers is the product 
of their moduli, and the amplitude of the product is the sum of their 
amplitudes. 

Example. Find the product of 3 (cis 30), 4(cis 150), and 
7(cis 72). 

Solution. The moduli of the given number are 4, 3, and 7. 
Hence in accordance with the theorem just stated the modulus 
of the product is 

4 X 3 X 7 = 84. 

The amplitudes of the given numbers are 30, 150, and 72. 
Hence, in accordance with the theorem just stated, the amplitude 
of the product is 

30 + 150 + 72 = 252. 
Therefore we have 

(3 cis 30) (4 cis 150) (7 cis 72) = 84(cis 252). 

88. The quotient of two complex numbers in polar form. To 

xu ^ 4. i*i(cos a + i sin a) . , , , ~ . 

express the quotient -~ ---- : ~ in the polar form we first 
r 2 (cos ft + i sm ft) ^ 

multiply both numerator and denominator by cos i sin ft 
and obtain 

ri(cos a + i sin a) (cos ft i sin ff) 
r 2 (cos ft + i sin ft) (cos ft i sin ft) 
or 

r\ f cos a cos ft + sin a sin ft + i(sin a cos ft cos a sin ft) "I 
r^ L cos 2 ft + sin- ft J" 



THE QUOTIENT Of TWO COMPLEX NUMBERS 207 

Using the subtraction formulas (10) and (11) of 53, we reduce 
this expression to 

- [cos (a - ft) + i sin (a - ft)]. 

7*2 

Using the notation cis for cos + i sin 0, we have 

or, stated in words: 

The modulus of the quotient of two complex numbers is the 
quotient of their moduli, and the amplitude of the quotient is the 
difference of their amplitudes. 

Evidently multiplication and division are very simply per- 
formed when the numbers are in the polar form. If the numbers 
are in the rectangular form a + bi and the amount of multiplica- 
tion and division involved is extensive, the numbers should be 
changed to the polar form and then combined in accordance with 
the theorems just stated. 

EXERCISES 

In this set of exercises give your results in the a + bi form. 

1. Perform the indicated operations: 

4(cos 27 + t sin 27)5(cos 34 + i sin 34). 

(6) 7(cis 129)4(cis311). 

( \ 6 cis 43 (d . 1 cis 143f 

( 2 cis 87' W 5 cis 17" 

2. Perform the indicated operations: 



3 + 3\/3* 

i). 



(d) (l 

co 



tf) (-2 + 20(3 - 3V3i). (j) 5(co 8 80+t8in80 ) > 

2 - ' 



208 COMPLEX NUMBERS [CHAP. X 

3. Perform the indicated operations: 

( v 7(cis 3Q)6(cis 45) 
W 2-2* 



(c) 



7 cis 150 
(A/2 - V2i)(3 - 3x/3t) 



(\/2 + \/2t)(cis 120) 
(d) (1 + t)(\/2 - V20*(3 + 3-V/303 cis 225. 



4. Perform the indicated operations. 



(5cis32) 5 (4ci84Q) 4 

(20 cis 10)* 
(5-2 -7.1t)(6.4 + 5.2t) 



8.3 + 4.6* 
(c) 7(cis 330)6(cis 1764). 

89. Powers and roots of complex numbers. De Moivre's 
theorem. If, in (2), all the values of r be taken equal to unity 
and all the angles equal to 6, we obtain (cis 0) n = cis nO, or 

(cos + i sin 0) n = cos nB + i sin nO. (4) 

This relation is known as De Moivre's theorem. Although wo 
have proved it only when n is an integer, it is true for all real 
values of n. 

Since the sine and the cosine of an angle are unchanged when 
the angle is changed by any multiple of 360, formula (4) holds 
true when is replaced by 

8 + 2fcir, or 6 + fc 360, k is an integer. (5) 

When n is an integer the addition of k 360 to gives rise to 
nothing new; but when n is fractional a number of values of 
cis (n0 + fcn360) may be found by assigning different values 
to k. Thus, to find the nth root of x + yi where n is an integer, 
write 

i^ i 

(x + yi)" = {r[cos (6 + fc360) + i sin (0 + k 360)] } 

k T (* . k 380\ ... (e . k 360\ 

= r n cos I -- I + i sin I -- I 

L \n n ) \n n / 



89] POWERS AND ROOTS OF COMPLEX NUMBERS 209 
or, using the notation cis for cos 6 + i sin 

, (6) 



where fc may be any integer. By letting k assume in succession 
the values 0, 1, 2, , n 1, we obtain from (6), n distinct 
results, that is, n distinct complex numbers, each one of which is 
an nth root of x + yi. If k be assigned an additional value, the 
amplitude of the resulting number will differ from the amplitude 
of one of the roots just found by a multiple of 2?r; that is, this 
new number will be equivalent to one of the roots already found. 
Also it can easily be proved that a complex number cannot have 
more than n different nth roots. Therefore, if n is an integer, 
every complex number different from zero has n and only n 
distinct nth roots. 

Example 1. Find the three cube roots of 8. 
Solution. Expressing the number 8 + Oi in the polar form 
and using the general value of the amplitude, we obtain 

-8 = 8 cis (180 + fc360) (a) 

Extracting the cube root of (a) and using (6), we obtain 



' . 7 

cis ( - + fc 



36 A 

I- 



3 

Giving k the values 0, 1, 2 in succession, we obtain 

2 cis TT = 2(-l + 00 = -2, 
2 cis y = 2(i - 1 -x/30 = 1 - iV3. 

Example 2. Find the four fourth roots of 3 + 3\/3i. 

Solution. Plotting the given number and solving the triangle 
exhibited in Pig. 5, we write direct from the figure the polar form 
of ( 3 + 3\/30> using the general value of the amplitude. 
This gives 

-3 + 3V3i = 6 cis (120 + fc360). 



210 



COMPLEX NUMBERS 



[CHAP. X 



Extracting the fourth root and using (6), we obtain 



(-3 



= 6* ci 



cis 



k 



= 1.565 cis (30 + 90). (a) 

Assigning to k in (a) the values 0, 1, 2, 3, we obtain as the 
roots of -3 + 3V& 

1.665 cis 30, 1.565 cis 120% 1.565 cis 210, 1.565 cis 300 

or in the a + bi form 

1.366 + 0.782i, -0.782 + 1.365i, -1.365 - 0.782i, 

0.782 - 1.3661 



P(-3,3V3) 




120 




Since the moduli of the roots are equal, the points representing 
these roots will be on the circumference of a circle (sec Fig. 0) 
having its radius equal to the common modulus of the roots and 
having its center at the origin. Since the amplitudes of any 
pair of successive roots differ by 360/n, the points representing 
the roots are equally spaced along the circumference of the circle*. 
Hence, after one root is located, it is easy to plot the remaining 
roots and to express them from the graph in the polar form. 

EXERCISES 

1. Find the values of each of the following numbers giving the results 
in polar form: 

(a) [2 cis 120]'. (d) (| + ^3i) 3 . 

(b) [4 cis I*-] 7 , (e) (3 - 3i) 6 . 

(c) (cis 10). (/) d + i)- 4 - 

2. Find the indicated roots, giving the results in polar form: 

(a) (10 - 6i)i. (e) (5.6 - 7.2i)t. 

(b) (i - f\/3iU (/) U4(cis 45 + *360)]l. 

(c) it. (g) [cis Or + 2for)]l. 

(d) (-l)i. 



EXPONENTIAL FORMS OF A COMPLEX NUMBER 211 

3. Solve the following equations: 

(a) x 3 + i = 0. (d) x 9 - 2* 3 - 35 = 0. 

(6) z 6 = -32. (e) x 1 - x 4 + x* - 1 = 0. 

4. Derive the formula for cos 20 and sin 26 by expanding the left-hand 
member of (cos + i sin d) 2 = cos 20 + i sin 20, and then equating the 
real parts and the imaginary parts of the two members. 

6. Using the formula (cos 8 + i sin 0) n = cos nd + i sin n0 and giv- 
ing n appropriate values, derive formulas for cos 30,* sin 30, cos 50, and 
sin 50. 

Hint. Letting n = 3, we have 

[cos + i sin 0] 3 = cos 30 + i sin 30 
or, expanding the loft-hand member, 

cos j f i 3 cos- sin 0-3 cos sin 2 + f sin 3 - cos 30 + i sin 30 
or 
(cos-* 0-3 cos sin 2 0) + i(3 cos 2 sin + sin 3 0) = cos 30 + i sin 30. 

Now equate the real part of the left-hand member of the above equation 
to the real part of the right-hand member to obtain the formula for 
cos 30. 

90. Exponential forms of a complex number. In higher 
mathematics we find justification for the equation 

r(eos + i sin 0) = re'*, (7) 

where is expressed in radians and c(= 2.71828, approximately) 
is the base of the system of natural logarithms. Thus we have 
another form in which to write a complex number. 
From (7) we write 

cos + i sin = e i9 , 
cos i sin = e~ l . 

Solving these simultaneously for cos and sin 0, we obtain 

( >i9 4- p-i8 p ie _ p-i8 

cos0 = V -' sin0 = 6 A~ (8) 



* This formula may be used to obtain an elegant solution of the cubic 
equation. 



212 COMPLEX NUMBERS [CHAP. X 

These relations were stated by Euler in 1743. Taking them 
as fundamental definitions and further defining tan 0, cot 6, 
sec 6, and esc by the equations 

. sin 6 , - cos . 1 - 

tan = - r; cot = . ^ sec = - ^ esc = 



. 
cos sin a cos 6 sin 

we may develop, independent of any geometric meaning attached 
to the functions or their arguments, all the formulas of trigo- 
nometry. It is also interesting to observe that the theorems 
relating to multiplication, division, involution, and evolution of 
complex numbers are easily proved by using this exponential 
form. 

EXERCISES 

-A 

1. Use (7) to evaluate e**, e 2 , e* 2 , e 4 . 

2. Use (8) to find cos 2i and sin 2i. 

k'* + 1 

3. Prove that cos (t log e k) = - sr 

4. Assume that (7) holds true, and use it to prove De Moivre's 
theorem. 

6. Use (8) to prove 

(a) cos 2 6 + sin 2 0=1. 

(b) cos (A + B) = cos A cos B sin A sin B. 

6. Use (7) to evaluate e (2 *+ 1)Tl , where k is an integer; then show that 
log. (-1) = (2* + !)'. 

91. The hyperbolic functions. A class of functions very useful 
in many fields is analogous to the trigonometric functions. 
The function cos id is called the hyperbolic cosine of B and is 
written cosh 6. Similarly i sin iB is called the hyperbolic 
sine of B and is written sinh B. Using (7) with B replaced by iA, 
cos iA by cosh A, and i sin iA by sinh A, we have 

g>A _|_ a A pA- o A 

cosh A = e \^ , sinh A = * 2 e (9) 

Corresponding to other trigonometric functions, there are four 
other hyperbolic functions defined as 



91] THE HYPERBOLIC FUNCTIONS 213 



. - A sinh A cosh A 1 

tann A = = ri coth A = . , . = - =~ -- 
cosh A sinh A tanh A 

sech A = = -r> csch A = 
cosh A 




(10) 



and named by prefixing the word hyperbolic to the names of 
their trigonometric counterparts. 

Example. Using the definitions (9), verify 

cosh 2 A sinh 2 A = 1. (a) 

Solution. From (9) 



cosh 2 A = - = e" + + J<r". (6) 

sinh 2 A = f?lI A Y = e" - + 6-". (c) 



Subtracting (c) from (6), member by member, we obtain 
cosh 2 A - sinh 2 A = 1. 

EXERCISES 

1. Find cosh 0, sinh 0, cosh 1, sinh 1. 

2. Prove that cosh x is always positive and greater than 1 if x is a 
real number. 

3. Prove that the value of tanh x is numerically less than 1 for all 
real values of x. What other hyperbolic function is always less than 1? 

4. Using definitions (9) and (10), show that 

sinh (x) = sinh x, cosh ( x) cosh x, *tanh ( x) = tanh x. 
6. Show that cosh x + sinh x = e x , cosh x sinh = e~ x . 
6. Using definitions (9) and (10), verify the following identities: 

(a) tanh 2 x + sech 2 x 1. 

(b) coth 2 x csch 2 x = 1. 

(c) sinh (x y) = sinh x cosh ?/ cosh x sinh y. 

(d) cosh (x y) = cosh x cosh y sinh x sinh y. 

^ 



(e) sinh x + sinh y = 2 sinh f ^ ) cos ^ ( 5 
(/) sinh x sinh y = 2 cosh f ^ ) sm ^ ( o / 
(g) cosh Z + cosh y = 2 cosh f ^ ) cos ^ ( 2 ) 

X ~\~ 11 X ~~~ t/ 

(h) cosh x cosh y = 2 sinh s s ^ nn o 



214 COMPLEX NUMBERS [CHAP. X 

7. In the equation 

x = sinh t/, (a) 

y is a number whose hyperbolic sine is x. We express this by writing 

y = sinh" 1 x, 



and define the symbol sinh~ l x to be the number whose hyperbolic sine is x. 

In equa 
ehow that 



e y e~ v 
In equation (a) replace sinh y by - ---, solve the result for y, and 



sinh" 1 x = log (x + \/x 2 + 1). 

8. The symbol cosh~ l x means the number whose hyperbolic cosine 
is x and is read the number whose hyperbolic cosine is x. The tanh- 1 jr t 
coth" 1 x, sech" 1 x, csch- 1 x are defined and read in an analogous manner. 

Proceed in a manner similar to that of problem (7) and show that 



cosh- 1 x = log (x + 



1, 1+x 
g log j-^r^' 



9. Show that 



sinh- 1 x csch- 1 -i 
x 

cosh" 1 x = sech" 1 - 

tanh" 1 x = coth" 1 
x 

92. MISCELLANEOUS EXERCISES 

1. Plot the following complex numbers and write them in the form 
x + yi: 

(a) 3 cis 45. (e) 5 cis 58. 

(b) 4 cis 150. (/) 8 cis 124. 

(c) 5 cis 300. (0) 6 cis 324. 

(d) 7 cis 90. (h) 2 cis 22020'. 

2. Write the following complex numbers in the polar form: 

(a) 2 + 2i. (d) 2 - 3i. (g) - 2i. 

(b) 3 - 3i. (e) -3 - 4i. (h) -3.2 - 2Ai. 

(c) -3+i. (/) 5 - 6t. (*) -4.2 + 1.4i 



92] MISCELLANEOUS EXERCISES 216 

3. Perform the indicated operations: 

(7 cis 45) (8 cis 300) 
(a) 4 cis 135 

,.. (8 cis 47) (3 cis 200) 
(6) Z" 



,,, 
(d) 



_ 

(7 - ttt)(4 - 

(8.2-3.41) (7.1 +3.K) 
- 



4. Find the values of each of the following numbers, giving; the results 
in polar form: 

(a) [2 cis 45] 5 . (c) (1 - \/3~0 4 . 

(b) [2.6 cis 73]-'. (d) (-3 + 40 B . 

6. Find the indicated roots, giving the results in polar form: 



(a) _ 

(b) \ft~^3i. ___ (e) V^l 

(c) >^~-3.4 - 5.1i. (/) v^3.~fi~+ JUtf. 

6. Solve the following equations : 

(a) s 8 - 8 = 0. (c) x 8 = 3 - 4i. 

(/>) j- 3 = i. (r/) u- 7 = -3.8 - 7i. 

7. Show that 

1 (e** e~ ix \ 
tan x = . l i~~~~ J- 
i \e ix + e~ lz / 

8. Prove that 

2e l * 
oc x = ?i , + - 1 - 

9. Using definitions (0) and (10), verify the following identities. 

, v . , f , , tanh a; + tanh ?/ 
(a) tanh (x + y) = Y+toi5iT- tanh y 

... A . , x tanh j; tanh ?/ 

(/>) tanh (.r y) , -r~ ,- ------ r~ i * 

v y v J/ 1 tanh x tanh y 

(c) sinh 2x = 2 sinh j: cosh a*. 

(d) cosh 2x = cosh 2 x + sinh 2 x = 2 cosh 2 x 1 = 1 + 2 sinh 2 j. 



CHAPTER XI 
LOGARITHMS 

93. Introduction. The labor involved in many numerical 
computations is considerably lessened by the use of logarithms. 
In the following articles we shall discover that in a sense the use 
of logarithms reduces multiplication to addition, division to 
subtraction, raising to a power to multiplication, and extracting 
a root to division. For this reason logarithms constitute a 
remarkable labor-saving device in computation. 

We shall learn presently that logarithms are exponents and 
that the laws that govern the use of exponents are the ones that 
govern the use of logarithms. Hence, before discussing loga- 
rithms, we shall recall from algebra the laws of exponents. 

94. Laws of exponents. It is proved in algebra that, when 
the exponents m and n are any numbers, the following laws hold: 

(I) a m a n = a m+n . (IV) (a6) m = a m b m . 



(Ill) (a m ) n = a mn . 

EXERCISES 

1. Evaluate the following : 

(a) 3 2 3~ 3 . (d) 3-33 I. (g) (25 X 49)-*. 



(b) 7-?^7". (e) ~^ (h) 

(c) 3-*3. (/) (3->). (t) 

2. Find, in each case, the value of x which satisfies the equation: 



(o) 10' = 1000. 
(6) 3-' = x. 
(c) x 4 = 10,000. 


(J) a;- 2 = 100. 
(g) 10 = x. 
(K) ar* = 10. 


(t) 7- = 1. 
(0 or 1 = 0.01. 
(m) 7 = 343. 


(d) x~\ = 3. 


(i) (36)' = . 


(n) (j)" 2 = 16 


W 4* =i . 


0') ,- = V7. 


i 
(o) 2 = 4. 



216 



95] DEFINITION OF A LOGARITHM 217 

3. Find x if 

(a) 10* = A- (d) 10* = 1000. 

(6) 10* = 0.001. (e) 10* = 1. 

(c) 10* = 0.0001. (/) 10* = 100,000. 

4. Solve each of the following equations for x: 

(a) (3)(2)* + 4 = 100. (g) (3*)(9) 2 * = 3~!. 

(6) 5*+ 3 - 5 2 * = 0. (h) (ft H = 5Vx. 

(c) (8)(2)* - 2<* = 0. (i) (AH = 2*-'. 

(d) (8)(3*) = (27)(2*). (j) (7* a - 1 )(49 1 -) = >/7. 

(6) (x - 2)o = z 2 + 1. (*) (Y)~* - 3 " 2 = 3 ~ 3 ' 

(/) 27* = 81. (Z) ^VxVx = 64. 

95. Definition of a logarithm. If 6, L, and AT are numbers 
such that b raised to the power L is equal to N, then L is called 
the logarithm of N to the base b. In symbols, if 

b L = N, then L = log 6 #. (1) 

Stated differently, the logarithm of a number to a given base is 
the power to which the base must be raised to produce the 
number. 

The two equations in (1) express the same relation between 
the base 6, the number N, and the logarithm L. The second one 
is read: L is the logarithm of N to the base 6. Also N is called 
the antilogarithm of L (or the number whose logarithm is L) to the 
base b. Since 5 2 = 25, 2 is the logarithm of 25 to the base 5, 
and 25 is the antilogarithm of 2 to the base 5. Similarly, we have 

10 3 = 1000, /. 3 = logic 1000; 

10~ 2 = 0.01, /. -2 = logio 0.01; 
3* = V3, ' i = Ig 3 V3- 

Since I* = 1 for all values of r, 1 cannot be used as a base for 
logarithms. Also a negative number is not used as base; for 
many real numbers would have imaginary logarithms to a nega- 
tive base. For example, if ( 3)* = 27, x is imaginary. 
Although any positive number different from 1 might be used as a 
base, 10 is often chosen for reasons that will appear as our study 
continues. 



218 LOGARITHMS [CHAP. XI 

EXERCISES 

Write each of the following exponential equations as a logarithmic 
equation : 

1. 2 4 = 16. 4. (i)- = 4. 7. 25-5 = i 

2. 10 2 = 100. 6. 8i = 4. 8. 10 = 1. 

3. A/100 = 10. 6. 10- 2 = 0.01. 9. 10- 3 = 0.001. 

Write each of the following equations as an exponential equation: 

10. Iog 2 8 = 3. 12. Iog 7 49 = 2. 14. log, i = -. 

11. Iog 5 1 = 0. 13. logio 0.1 = -1. 16. Iog 9 1 = 0. 

In each of the following exercises, find the value of x: 

16. Iog 6 x = 2. 23. logio 100 = x. 30. log, 4!) = 2. 

17. log, 1 = 2. 24. Iog 2 32 = x. 31. Iog 27 3 = x. 

18. logs 25 = x. 25. Iog 5 (m) = * 32. Iog 2 (Tr:; = * 



19. log, 15 = 1. 26. logio x = 2. 33. logs x = 1. 

20. Iog 2 x = 3. 27. logio x = 2. 34. log*, x = 1. 

21. Iog 2 x = -2. 28. log, 3 = -. 36. log, () = 2. 

22. Iog 4 a; = -. 29. log, 49 = -2. 36. Iog 6 a; = 0. 

Show that 

37. (log* a) (log. 6) = 1. 

38. (Iogba)(logc6)(logac) = 1. 



39. log, = -1. 

40. Why cannot unity be used as a base for a system of logarithms? 

41. Why cannot a negative number be used as a base for a system of 
logarithms? 

96. Laws of logarithms. There are three fundamental laws of 
logarithms with which the student must be thoroughly familiar. 
These laws are easily derived from the laws of exponents. 

I. The logarithm of the product of two numbers is equal to the 
sum of the logarithms of the factors. 

Proof. Let M and N be any two positive numbers, and let 

x = logb N, and y = \og b M. (2) 



LAWS OF LOGARITHMS 219 

Then we may write 

fc* = N, and ^ = M. (3) 

Multiplying member by member the first of equations (3) by 
the second, we get 

b*b* = b*+* = MN, or Iog6 MN = x + y. (4) 
Substituting the values of x and y from (2) in (4), we get 
MN = log b M + log b N. 



By repeated application of the first law it is readily proved that 
the logarithm of the product of any finite number of factors is equal 
to the sum of the logarithms of the factors. 

II. The logarithm of a quotient is equal to the logarithm of the 
dividend minus the logarithm of the divisor. 

Proof. Dividing member by member the firsl <tf equations 
(3) by the second, we get 

\T fox \T 

= h f -v or loffi. = r // (%\ 

M fry - , 01 10g b M JT I/. ^) 

Substituting the values of JT and y from (2) in (5), we get 
log/, -^ = log ft N - log/, M. 

III. The logarithm of a number affected by an exponent is the 
product of the exponent and the logarithm of the number. 

Proof. Let 

x = Iog 6 N, or N = b*. (6) 

Raising both members of N = b x to the pt\\ power, we obtain 



Therefore, in accordance with (I) 

Iog 6 N p = px. (7) 

Substitution of the value of x from (6) in (7) gives 
log* NP = p Iog 6 N. 

Example 1. Find the value of log \/0.001. 
Solution, log VO.Wl = Iog 10 (0.001)^ = % Iog 10 0.001 

= i logio TWO = i(~3) = -f . 



220 



LOGARITHMS 



[CHAP. XI 



Solution, log 



Write Iog 6 
:,^ 



C " ~ C 

= Pog 4 a 2 + log* (c + d) 
[2 log* o + log* (c + d) - 5 log* c]. 



in expanded form. 

a 2 (c + d)* 
c 1 

log& c 6 ! 



Example 3. Write f log* (x + 1) + log* a; - 2 log* (a; 2 + 1) 

contracted form. 

Solution. | Iog 6 (a; + 1) + % log fc x - 2 log,, (z 2 + 1) 

= log* (x + l)i + log a:* - Iog 6 (* + I) 2 
. (x + 1)*** 



Another form of the answer is found as follows: 

(X + 1)W _ . 



log* 



+ I) 2 



1. Verify the following: 



EXERCISES 



1. 



(a) logio VlOOO + logic 

(b) Iog 2 VS + Iog 2 \/2 = 2. 

(c) log8(2)^ + log 7 (A)t= L. 

W iog 2 Vs + iog 3 (i) 2 =_j:^. 

(e) logs \/125 + logu Al/169 = Y- 

(/) logn rr + 2 logn \/n = 0. 

(g) Iog 2 (0.5) 8 - Iog 4 ^64 = -t. 

(^) Iog 6 1 - Iog 7 6 = 0. 

(i) Iog 10 10 5 - logio 10 2 + logio 10- 2 + logio 1 = 1. 

2. Write the following logarithmic expressions in expanded form: 



(o) logi -- 

(V) log, 

(c) logi 

(d) 



*.y 

>2 / 




y) 



, 

log6 



/,x 

(0 



97] COMMON LOGARITHMS. CHARACTERISTIC 221 

3. Write the following expressions in contracted form. 

(a) logb a + 2 logb c | logb d. 

(b) i logb a 3 logb c 4 log b (a + c). 

(c) i logb (a + c) + -3- log b (a c). 

(d) logb 3c -j- logb d + logb c. 

(e) |[logb a + 2 logb (c d) 4 logb c - i log fc (2 - a)]. 
(/) 5[ logb (a c) + logb (a + d) 6 logb d 2 log& a]. 

4. Take from a five-place table the following logarithms: 

logio 2 = 0.30103, logio 3 = 0.47712, logio 7 = 0.84510. 

From these numbers find logio 4, logio 9, logio 28, logio 32, logio 7, logio f . 

6. Using the logarithms in Exercise 4, find logio 1, logio f , logio 343, 
logio V2, logio "V 7 ?, logio 5. 

6. Using the logarithms in Exercise 4, find the value of the logarithm 
of each of the following expressions: 



(6) 

, . 
(c) 




97. Common logarithms. Characteristic. In computation, 
it is convenient and customary to employ logarithms to the 
base 10. Logarithms to this base are called common logarithms. 
Throughout this text we shall use common logarithms only, 
and we shall write log JV as an abbreviation of logio N. Thus 
when the base is omitted it will be understood that the base 
is 10. 

In this system of logarithms, the logarithm of any integral 
power of 10 is an integer, while the logarithm of any positive 
number not an integral power of 10 may be written as an integer 
plus a decimal. In general, the logarithm of a number consists 
of two parts, an integer called the characteristic, and a decimal 
called the mantissa. The characteristic is found by inspection; 
the mantissa is found from a table. We shall now deduce rules 
for finding the characteristic. 



222 



LOGARITHMS 



[CHAP. XI 



Consider the following table: 



10 5 = 100,000 


or 


log 100,000 = 5, 


10 4 = 10,000 


or 


log 10,000 = 4, 


10* = 1000 


or 


log 1000 = 3, 


10 2 = 100 


or 


log 100 = 2, 


10 1 = 10 


or 


log 10 = 1, 


10 = 1 


or 


log 1 = 0, 


10- * = 0.1 


or 


log 0.1 = -1, 


io- 2 = o.oi 


or 


log 0.01 - -2, 


10-3 = o 001 


or 


log 0.001 = -3, 


10" 4 = 0.0001 


or 


log 0.0001 = -4, 


io- 5 = o.ooooi 


or 


log 0.00001 = -5. 



From the foregoing table, we get by inspection the following 
information: 



Number 


Number of digits to 
left of decimal point 


Logarithm 


Characteristic 


1 < N < 10 


1 


-f- a decimal 





10 < N < 100 


2 


I -h a decimal 


1 


100 < N < 1000 


3 


2 + a decimal 


2 


1000 < N < 10,000 


4 


3 H- a decimal 


3 


10 < N < 10 fl 


n + 1 


n + a decimal 


n 



From the data just tabulated, we infer the following rule: 

Rule 1. The characteristic of the common logarithm of a number 
greater than 1 is positive and is one less than the number of digits 
to the left of the decimal point. 

Similarly, we get 





Number 








of zeros to 






Number 


right of 


Logarithm 


Characteristic 




decimal 








point 






0.1 <N<1 





1 + a decimal 


-1 or 9 - 10 


0.01 <JV<0.1 


1 


2 -f a decimal 


-2 or 8 - 10 


0.001 <N <0.01 


2 


3 + a decimal 


-3 or 7 - 10 


10- < N < lO-^- 1 ) 


n - 1 


n + a decimal 


-nor (10 - n) - 10 



98| EFFECT OF CHANGING THE DECIMAL POINT 

From the tabulated data, we infer the following rule: 



223 



Rule 2. The characteristic of the common logarithm of a positive 
number less than 1 is negative and is numerically one greater than 
the number of zeros immediately following the decimal point. 

When the characteristic is negative, it is convenient to add 10 
to the characteristic and subtract 10 at the right of the mantissa. 
Thus log 0.02545 = -2 + a decimal = 8 + a decimal 10. 
In general, if the characteristic n of log N is negative, change 
it to the equivalent value (10 n) 10, or (20 n) 20, 
etc. To obtain directly the characteristic of the logarithm of a 
number less than 1, subtract from 9 the number of zeros immediately 
following the decimal point; write the result before the mantissa 
and 10 after it. 

Illustrations: 



Number 


Characteristic 


Rule 


4261 


3 


1 


3.6121 





1 


0.1210 


-1 or 9 - 10 


2 


0.0025 


-3 or 7 - 10 


2 


0.00000345 


-6 or 4 - 10 


2 



EXERCISES 

Write the characteristic of the logarithm of each number: 

1. 7.613. 6. 761.3. 9. 89,261. 13. 3101. 

2. 467,916. 6. 31.12. 10. 412.16. 14. 14,481.10. 

3. 20.02. 7. 0.0371. 11. 0.0000309. 16. 0.30001. 

4. 3.00008. 8. 0.81219. 12. 0.003872. 16. 0.000810. 

98. Effect of changing the decimal point in a number. Any 

number may be written in the form N X 10*, where N is a num- 
ber between 1 and 10 and k is an integer. Thus we may write 
1,782,500 = 1.7825 X 10 6 , 17825 = 1.7825 X 10 4 . Evidently a 
shift of the decimal point appears in this notation as a change in 
k. Now log [N X 10*] = log N + k X 1. Since a shift of the 
decimal point changes ft, but not log N, it appears that the 
mantissa of log N is not affected by the position of the decimal 
point. In other words, a change in the position of the decimal 



224 LOGARITHMS [CHAP. XI 

point in a given sequence of figures has no effect on the mantissa; 
its sole effect is to change the characteristic. Because of this 
fact, 10 affords a particularly convenient base for a system of 
logarithms to be used for purposes of computation. 

99. The mantissa. Mantissas can be computed by use of 
advanced mathematics and, except in special cases, are unending 
decimal fractions. Computed mantissas are tabulated in tables 
of logarithms, also called tables of mantissas. These tables are 
called " three-place," "four-place," "five-place," etc., according 
as the mantissas tabulated contain 3, 4, 5, etc., significant figures. 
The choice of a table of logarithms should depend upon the degree 
of accuracy required and the accuracy of the data. In this text 
we shall discuss and use a five-place table, thus obtaining'results 
accurate to five significant figures. 

100. To find the logarithm of a number. In general, a five- 
place table of logarithms gives the mantissas of all integral num- 
bers lying between 999 and 10,000. The first three digits of 
the numbers are found in the left-hand column headed N, and the 
fourth digit is in the row at the top of the page. Therefore the 
mantissa of a number with four significant figures is in the row 
with the first three significant figures of the number and in the 
column headed by the fourth. 

Example 1. Find log 42.43. 

Solution. By the rule in 97, the characteristic is found to 
be 1. To find the mantissa, first find 424 in the left-hand column 
headed AT, then follow the row containing 424 until the column 
headed by 3 is reached. Here we find 62767. Therefore the 
mantissa is 0.62767. Hence 

log 42.43 = 1.62767. 

Example 2. Find log 0.0416. 

Solution. By the rule in 97, the characteristic is found to be 
8. -10. Using 4160, we find the mantissa to be 0.61909. 
Therefore 

log 0.0416 = 8.61909 - 10. 



101] INTERPOLATION 225 

EXERCISES 

Verify the following: 

1. log 2934 = 3.46746. 6. log 0.3132 = 9.49582 - 10. 

2. log 3.478 = 0.54133. 7. log 0.0003146 = 6.49776 - 10. 

3. log 28.7 = 1.45788. 8. log 0.03426 = 8.53479 - 10. 

4. log 1.817 = 0.25935. 9. log 0.272 = 9.43457 - 10. 

6. log 981.7 = 2.99198. 10. log 0.005075 = 7.70544 - 10. 

101. Interpolation. From the five-place table of logarithms we 
cannot obtain directly the logarithm of a number with five 
significant figures. However, by a process known as interpo- 
lation, we can find the mantissa of a number having a fifth 
significant figure. In this process we use the principle of pro- 
portional parts, which states that, for small changes in N, the 
corresponding changes in log N are proportional to the changes 
in N. Although this principle is not strictly true, it is sufficiently 
accurate to lead to results correct to the number of figures given 
in the table. 

The process of interpolation is illustrated by means of the 
following example: 

Example. Find log 235.47. 

Solution. From the table of logarithms we find the logarithms 
in the following form and then compute the differences exhibited. 

log 235.40 ) ) = 2.37181 ) ) 

log 235.47 / ' > 10 = ? / > 18 (tabular difference) 

log 235.50 ) = 2.37199 ) 

By the principle of proportional parts, we have 

is - is' or d - (IB) (18) = 13 (nearly) - 

We add d = 13 to the last two figures of 2.37181 to obtain 
log 235.47 = 2.37194. 

Notice that the value used for d was 13 instead of 12.6 because 
the table of logarithms is accurate only to five decimal places. 



226 LOGARITHMS [CHAP. XI 

In order to save work in interpolating when finding the man-, 
tissas of five-place numbers, each tabular difference occurring 
in the table has been multiplied by 0.1, 0.2, . . . 0.9, and the 
results placed on the right-hand sides of the pages when* these 
tabular differences occur. These tabulated results, called 
tables of proportional parts (P.P.), are headed by the tabular 
difference for which they have been formed, and the decimal 
points have been omitted. To interpolate in the example just 
solved, we locate the proportional parts table headed 18, and 
opposite 7 in the left-hand column we find d 13. 

EXERCISES 

Find the logarithm of each of the following: 

1. 40.488. 6. 0.0038345. 

2. 3.0473. 7. 0.080452. 

3. 10,201. 8. 0.000070123. 

4. 108.17. 9. 0.027038. 
6. 0.21544. 10. 0.18253. 

102. To find the number corresponding to a given logarithm. 

Generally in every problem involving logarithms, it is necessary 
not only to find the logarithms of numbers but also to perform 
the inverse process, that of finding a number corresponding to a 
given logarithm. 

If log N L, then N is the number corresponding to the 
logarithm L. The number N is called the antilogarithm of 
L. To find the antilogarithm N of the logarithm L, first use the 
given mantissa to find the sequence of figures in N, and then use 
the given characteristic to place 1 the decimal point so as to agree 
with the rule of 97. 

Example. Given log N = 1.60334, find N. 

Solution. The mantissa .60334 is not found exactly in the 
table, but we find the two successive mantissas .60325 and 
.60336, between which the given mantissa lies. From the table 
we find the numbers in the following form and then compute 
the differences exhibited. 

1.60325) ) = log 40.110) ) 
1.60334 J > 11 = log N ) X > 10 
1.60336 ) = log 40. 120 J 



103] THE USE OF LOGARITHMS IN COMPUTATIONS 227 

By the principle of proportional parts, we have 

x 9 (9) (10) Q , . , 

jg = jj> or x = ^ = 8 (nearly). 

We add x = 8 to the last figure of 40.110 to obtain 

N = 40.118. 

This interpolation should be performed by means of the table 
of proportional parts. In the P.P. column under the block 
corresponding to the tabular difference 11, we find the difference 9; 
immediately to the left of this w.e find 8, the fifth significant 
figure in the number N. 

EXERCISES 

Find x in each of the following: 

1. log x = 8.60200 - 10. 6. log x = 2.99876. 

2. log x = 3.89779. 7. log x = 0.87484. 

3. log x = 5.31664. 8. log x = 0.42239. 

4. log x = 9.70000 - 10. 9. log x = 1.11240. 

6. log x = 7.97295 - 10. 10. log x = 6.54782 - 10. 

11. Find x in each of the following: 

(a) logo; = -0.34345. (c) log x = -3.12864. 

(b) logs = -2.41325. (d) log x = -0.16132. 

103. The use of logarithms in computations. The following 
examples will illustrate how logarithms are used. 

Example 1. Evaluate (461) (4.321). 

Solution. Denoting the product by jc, we may write 

x = (461)(4.321). 

Equating the logarithms of the two members of this equation, 
we get 

log x = log 461 + log 4.321. 

Looking up the logarithms of the numbers, we obtain 

log 461 = 2.66370 

log 4.321 = 0.63558 

Adding, we have log x = 3.29928. 



228 LOGARITHMS [CHAP. XI 

The antilogarithm of 3.29928, is 

x = 1992.0. 

Example2. Evaluate 

v 7 ,- T , (217) (3. 18) 

Solution. Let x = 



Then log x = log 217 + log 3.18 - log 62.142. 

log 217 = 2.33646 

log 3. 18 = 0.50243 

Sum = 2.83889 

log 62.142 = 1.79338 

Subtracting, we obtain log x = 1.04551 

The antilogarithm of 1.04551 is 

x = 11.106. 

Example 3. Evaluate (2.7 13) 3 . 
Solution. Lets = (2.713) 3 . Then 

log x = 3 log 2.713 = 3(0.43345) = 1.30035. 
/. x = 19.969. 

Example 4. Evaluate y^O.7214. 

Solution. Lets = v^O.7214 = (0.7214)*. Then 

log re = ^ log 0.7214 = ^(9.85818 - 10). 

If we should divide this logarithm by 3, the characteristic of the 
resulting logarithm would not be in the standard form. Hence 
we first add 20 and then subtract 20, writing the logarithm in 
the form 29.85818 30. Then we write 

3)29.85818 - 30 

Dividing, we get log x = 9.95273 - 10 
or x = 0.89688. 



104] COLOOARITHMS 229 

EXERCISES 

Evaluate the following: 

1. 52,564 X 0.0082546. 4. 7*. 7. (33.982)-*. 

0.0031593X684.82 , 75,859 X 0.0028242 

2 - "0.0096548 - 6 ' (0 ' 03628) ^ 8 ' -37^68 X 0.09185 ' 

3. (1.045)". 6. V(442.84) 3 . 

104. Cologarithms. Subtracting a first number from a second 
is equivalent to adding the negative of the first to the second. 
Hence, to avoid subtraction in dealing with logarithms, we 
introduce cologarithms. 

The cologarithm of a number is the negative of its logarithm. 
Therefore adding the cologarithm of a number is equivalent to 
subtracting its logarithm. 

To avoid negative mantissas, the cologarithm of a number w, 
written colog n, is found by using the form colog n 10 
logn - 10. Thus colog 2 = 10 - log 2 - 10 = 10 - 0.30103 - 
10 = 9.69897 - 10, and colog 0.3 = 10 - (9.47712 - 10) - 
10 = 0.52288. The student will find it convenient in getting 
colog n to begin at the left of log n, subtract each of its digits from 
9 except the last significant one, and subtract that from 10. 

The following example will illustrate the use of cologarithms. 

Example. Find x if x = 



Solution. log x = log 342.10 - log 6710 - log 0.31820 

= log 342.10 + colog 6710 + colog 0.31820 

log 342.10 = 2.53415 

log 6710 = 3.82672, colog 6710 = 6.17328 - 10 

log 0.31820 = 9.50270 - 10, colog 0.31820 = 0.49730 

log x = 9.20473 - 10 
and x = 0.16023. 

EXERCISES 
1. Verify the following: 

(a) colog 179.82 = 7.74516 - 10. 

(6) colog 0.63273 = 0.19878. 

(c) colog 7.5328 = 9.12304 - 10. 

(d) colog 23.975 = 8.62024 - 10. 



230 LOGARITHMS [CHAP. XI 

2. Using cologarithms, find the value of 

f \ 36 - 21 ,M 42.21 4L262 142.3 

W 7.215' W 0.2861' lc; (61. 84) (1612)' w o!02813' 

105. Computation by logarithms. In solving complicated 
problems, the computer is helped materially by a good form. 
The one discussed below has the advantages of simplicity, 
completeness of record, and brevity. It is practically self- 
explanatory since the main feature consists in reference of every 
function on a line to the first number in the line; a complete 
record of logarithms and operations is tabulated, and little 
writing is required. Since the outline of the form can always 
be made in advance, the student should first make this outline 
and then perform the computation without interruption. Speed 
and accuracy are gained by this method. 

The form will be used in the following solution. 

fli \/br' 2 
Example 1. Find x if x = Y 4 and a = 8.1632, b = 

de* 

729.77, c = 0.46330, d = 5.2133, c = 0.32411. 
Solution. First write tin* formula 

log x = -J- log a + -J- log b + 2 log c + colog d + 4 colog e. 
The following form contains the solution: 



a = 8.1632 
b = 729.77 
c = 0.04633 
d = 5.2133 
e = 0.32411 



log a = 0.91186 

log b = 2.86318 

log c = 8.66586 - 10 

logd = 0.71711 

log e = 9.51069 - 10 



S- log a = 0.30395 
y log 6 = 0.57264 
2 log c = 7.33172 - 10 
colog d = 9.28289 - 10 
4 colog e = 1.95724 



x = 0.28083 log x = 9.44844 - 10 

Note that each number in any line relates to the first number 
in the line, and the relation is indicated that the record of oper- 
ations is complete, that little writing is required, and that an 
examiner could easily perceive and follow the steps taken. 

In the following solution a form is indicated, but the computa- 
tion is left as in exercise to the student. 



Example 2. Find x if x = |vc X a 2 1 where fl = 61 214 

L a + Ve J 



106] REMARKS ON COMPUTATION BY LOGARITHMS 231 

c 

+ 
c = 12.112, ande = 139.02. 

Solution. First we write the formula 

log x = ^ log c + 2 log a + colog (a + 
and then make the following form: 



: + 



139.02 



a = 61.214 



12.112 



log e = log e 



log a = 
log (a -f 
log c = 



log \/e 



2 log a 

colog (a -f \/e) 

ijog c _____ 

~ 



log x 



The student should perform the computation to obtain x = 
5.6319. 

EXERCISES 

Make a form or outline for computing each of the following: 



l (32.861) 2 (3.1416)* 



(62.181) 3 L rf 5 e 

3 [ 'n.64) 2 (C>2.T2) A 



/(31. 

- ^ 



106. Remarks on computation by logarithms. 

(a) When interpolating, do not carry logarithms beyond the num- 
ber of decimal places given in the table used. 

(b) When evaluating an expression containing negative numbers, 
use logarithms to compute desired positive components, and then 
combine the results with appropriate signs. In this text a symbol 
( ) before a logarithm will indicate that a negative number is under 
consideration; thus if log x = (-)9.87123 - 10, x = -0.74342.* 

(c) Make a form like that of Example 1, 105, before beginning 
computation. 

(d) Strive for accuracy in computation. Speed comes with 
practice. 

* This does not moan that a negative number has a real logarithm. The 
minus symbols serve merely to keep a record of the signs involved in the 
given expression. 



232 



LOGARITHMS 



[CHAP. XI 



Example. Find the value of x if x 



/(-47.123) 2 (-36.184)* 
^ V3L118 



Solution. 
log (-x) : 



log 47.123 + log 36.184 + colog 31.118]. 



o = 

6 = 

c = 



47.123 
36.184 
31.118 



loga = (-H.67324 
log b = (-)l. 55852 
log c = 1.49301 



2 log a = 3.34648 
| log b = (-)0.51951 
colog c = 9.25350 - 10 



x = -4.2063 



5) (-)3. 11949 
logo; = (-)0.62390 



EXERCISES 

Find by use of logarithms the results of the following exercises. In 
each case make a complete outline or form before using the tables. 



1. 3.1416 X 2.7183. 

2. 29.572 X 0.00036841. 

3. 335,000,000 X 0.000099854. 

4. 2727.5 X 0.37375. 

6. 1487 X 3.139 X 42.96. 
g 2.9275 X 34.278 
505.92 

48.962 X 39.595 

78.545 
g 2964.5 X 38.423 
75.65 X 84.384 * 
9 2954.5 X 64.532 
" ~91~1.36 X 318.5 " 

10 26.893 X 0.0000545 
319.62 X 0.00068432" 

11. (1.5) 16 . 



14. A^O. 17638 X 2.1279. 
15. 



38.345J 

16. (0.00062584)*. 

17. (665.35)-K 



18. 



.45) (423.34) 



7. 



(178) (89) 

19 (-80,941)^-0.031. 
(54,082)^00712 

4 X 28.7 X V345 
29 X 137 

21. V(67.811) 2 + (~ 



20. 



23. 



12. 

24. [(-8.90172)(732.95)*(0.0954)*] 2 . 



26. V(27.5) 2 - (3.4S3) 2 .* 



VT7631.25) 2 - (6712.15) 2 .* 

.?f(23.975)(5.793) 2 
^ 179.82 



5086(-0.0008769) 8 



(9802)(0.001984) 4 



* Hint. First factor the radicand. 



106] REMARKS ON COMPUTATION BY LOGARITHMS 233 
1954.7 X -^0030121 



27. 4/ 

^17,959 X (0.84132) 8 (560.63) 

28 (0.04)1(0.057897)* 

(87.67) - 9 

29 */ (348.7) 2 (-2.685) 3 (3.08'212r 

' \(2.678)i(0.08216) 4 (-800,013)' 

30 3 /(Q-QQ2452)I(86.47) 3 (- 128721) 
*'(-5280)(-0.071~i5) 2 (-62.472)' 

3.r 
31. J-p^rt a = 7.5328, 6 = 6384. 




32. . /^ - Va 2 ^; a = 735.9, 6 = 0.198, c = 27. 

^a 3 

33. a ^-; /) = a + c 2 , a = 23.722, 6 = 571.17, c = 0.03218. 

34. Given a = 3.7124, 6 = 32.617, find log (a + 6), log (a - . 



log -r- 



35. Find 1C, given s = i(a + 6 + c + d), 



K = V(s- a)(s - b)(s - e)(a - d), 
a = 6.3246, 6 = 7.7459, c = 8.5441, d = 5.1961. 

36. a - l ^> given a = 0.00275, b = 100.5, c = 5075.5, d = 0.001875. 



37. , given a = 30 i.03, b = 0.00036954, c = 0.0028182, 
L e 2 / 3 ^ 4 J 

d = 35,890,000, e = 0.000002814, / = 561.29, g = 2718.3. 

38. Find the weight of a steel sphere 1.0127 ft. in diameter if steel 
weighs 490 Ib. per cu. ft. 

39. Find the weight of a cube of metal weighing 530 Ib. per cu. ft. 
if the edge of the cube is 1.6271 ft. 

40. A conical piece of wood weighs 92 Ib. If the area of the base of 
the solid is 1.3341 sq. ft., find the altitude. (The wood weighs 33 Ib. 
per cu. ft.) 

41. During a rain 0.521 in. of water fell. Find how many gallons of 
water fell on a level 10.7-acre park. (Take 1 cu. ft. = 7.48 gal., 
1 acre = 43,560 sq. ft.) 

42. The time t of oscillation of a simple pendulum of length I ft. is 
given in seconds by the formula 



234 LOGARITHMS [CHAP. XI 

t = 



32.16 

Find the time of oscillation of a pendulum 3.326 ft. long. (Take 
TT = 3.142.) 

43. What is the weight in tons of a solid cast-iron sphere whose 
radius is 5.343 ft. if the weight of 1 cu. ft. of water is 62.355 Ib. and the 
specific gravity of cast iron is 7.154? 

44. Find the volume and surface of a sphere of radius 14.71. 

45. The stretch of a brass wire when a weight is hung at its free end 
is given by the relation 



where m is the weight applied, g = 980, I is the length of the wire, r is 
its radius, and A; is a constant. Find k for the following values: m =- 
944.2 g., I = 219.2 cm., r = 0.32 cm., and 8 = 0.060 cm. 

46. Find the length I of a wire that stretches 5.9 cm. for a weight of 
1826.5 g. hanging at its free end, when the diameter of the wire is 
0.064 cm. and k = 1.1 X 10 12 . 

47. The weight P in pounds that will crush a solid cylindrical cast- 
iron column is given by the formula 

,73.56 

P = 98,920 -jpp 

where d is the diameter in inches and I the length in feet. What weight 
will crush a cast-iron column 6 ft. long and 4.3 in. in diameter? 

48. For wrought-iron columns the crushing weight is given by 

d 355 
P = 299,600 -p-. 

What weight will crush a wrought-iron column of the same dimensions 
as that in Problem 47? 

49. The weight W of 1 cu. ft. of saturated steam depends upon the 
pressure in the boiler according to the formula 

PO 941 
TXT _ _ .. 

w ~ 330.36 

where P is the pressure in pounds per square inch. What is W if the 
pressure is 280 Ib. per sq. in.? 



108] EQUATIONS OF THE FORM x = o\ a = & 235 

107. Change of base in logarithms. Occasionally it is neces- 
sary to find the logarithm of a number AT to a base 6 other than 
10. To do this we let 

Iog6 N = x, or b* = N. 

Equating the logarithms to the base 10 of the two members of 
this equation, we get 

1 t 1 AT l 

x logic b = log Jv, or x = -, 



-, r 
log 10 o 

Since the divisor and dividend of this fraction are logarithms, 
they will generally bo numbers of several digits. Therefore 
it is advisable to perform the indicated division by means of 
logarithms. 

Example. Find the value of Iog 3 0.092118. 

Solution. Let x = log, 0.092118. Then 3* = 0.092118. 
Equating the logarithms to the base 10 of the two members of 
this equation, we obtain 

x log 3 = log 0.092118 
or 

= logioJ^092U8 8.96434 - 10 = - 1.03566 
X ~~ logio 3 "" 0.47712 0.47712" 

This quotient is evaluated as follows: 

a = - L.0357 I log a = (-)0.01523 

b = 0.47712 | log 6 = 9.67863 - 10 | colog b = 0.32137 
x = -2.1707 log* = (~)0.33660 

108. Solution of equations of the form x = a b y a = x b . We 

shall now illustrate the method of solving equations of the 
form x = a b , and a = x b , in which a and b are given numbers. 

Example 1. Find x if x = (3.21) 8 - 27 . 

Solution, log x = 8.27 log 3.21 = (8.27) (0.50651). 

The solution is displayed below. 



a = 8.27 
b = 0.50651 



log a = 0.91751 

log 6 = 9.70459 - 10 



log x = 4.1889 log (log x) = 0.62210 



236 LOGARITHMS [CHAP. XI 

Therefore log x = 4.1889, from which we get x = 15,449. 

Example 2. Find x if z 7 - 2143 = 0.080133. 
Solution. Equate the logarithms of the two members of the 
given equation and solve for log x to obtain 

7.2143 log x = log 0.080133 
or 

_ log 0.080133 8.90381 - 10 = -1.09619 
log x - 7 2143 - 7.2143 " 7.2143 

The evaluation of the quotient for log x follows: 

a= -1.0962 I log a = (-)0.03989 

b = 7.2143 |log b = 0.85820 [cplog b = 9.14180-10 

log x = -0.15195 log"(log x) = (-79.18169- 10 

To make the mantissa of log x positive add it to 10 10 to 
obtain 

log x = 10 - 0.15195 - 10 = 9.84805 - 10. 

Therefore 

x = 0.70477. 

EXERCISES 

1. 

1. x = Iog 7 100. 9. 5* = 1.307. 

2. x = logo.ss 99,324. 10. 5 2 * = 317.46. 

3. x = Iog 27 0.00328. 11. log* 8 = 0.35678. 

4. x = logo.0964 87.543. 12. log, 2 = 0.69315. 

5. x = Iog 20 100. 13. log, 0.07936 = 2.983. 

6. x = logs 27,569. 14. x 2 - 892 = 0.07936. 

j. 

7. x = log,. 7 0.8173. 16. (1.5)* = 32. 

i 

8. x = Iog 2l 0.09827. 16. 4.02 = (2.37)*+ x . 

17. Given 3*+* = 2(5), x - y = 1, find x and y. 

18. How long will it take a sum of money to double itself if put at 
4 per cent compound interest? This is represented by (1.04)* = 2 
where x is the number of years. Solve for x. 

19. Solve the equation e* + er* = y, for x (a) when y = 2, (6) when 
y = 4. e = 2.7183. 



109] 



GRAPH OF y 



237 



20. If fluid friction is used to retard the motion of a flywheel making 
Fo revolutions per min., the formula V = Vye~ kt gives the number' of 
revolutions per minute after the friction has been applied t seconds. 
If the constant k = 0.35, how long must the friction be applied to reduce 
the number of revolutions from 200 to 50 per min.? e = 2.7183. 

21. The pressure, P, of the atmosphere in pounds per square inch, 
at a height of z ft. is given approximately by the relation 



where Po is the pressure at sea level and A: is a constant. Observations 
at sea level give P = 14.72, and at a height of 1122 ft., P = 14*11. 
What is the value of k? 

22. Assuming the law in Exercise 21 to hold, at what height will the 
pressure be half as great as at sea level? 

23. If a body of temperature r l\ is surrounded by cooler air of tem- 
perature TV, the body will gradually become cooler, and its tempera- 
ture, T, after a certain time, say t min., is given by Newton's law of 
cooling, that is, 

T = To + (Ti - To)*-*', 

where A; is a constant. In an experiment a body of temperature 55C. 
was left to itself in air whose temperature was 15C. After 11 min. 
the temperature was found to be 25. What is the value of &? 

24. Assuming the value of k found in Exercise 23, what time will 
elapse before the temperature of the body drops from 25 to 20? 

26. Solve the equation log. (3x + 1) = 2 for x. 
26. Solve the equation logic (x 2 2lx) = 2 for x. 

109. Graph of y = logio x. If wo assign values to x in the 
equation y = logio x and find the corresponding values of y, 
we shall obtain the coordinates of points on the curve y = logio x. 
A few of these values are tabulated in the accompanying table. 
Plotting these points and drawing a smooth curve through 



X 


0.5 


1 


3 

0.48 


5 


8 


10 


15 


20 


25 


30 


35 


40 


y 


-0.3 





0.70 


0.9 


1 


1.17 


1.3 


1.4 


1.48 


1.54 


1.6 



them, we obtain the graph shown in Fig. 1. For convenience, 
the unit on the y-axis has been taken ten times as large as the 
unit on the z-axis. 



238 



LOGARITHMS 



[CHAP. XI 



If the student retains a mental picture of this graph, he will 
find it easy to recall the following facts: 

(a) A negative number has no real number for its logarithm. 

(6) The logarithm of a positive number is negative or positive 
according as the number is less than or greater than 1 . 

(c) If the number x approaches zero, log or decreases without 
limit. 

(d) If the number x increases indefinitely, log x increases 
without limit. 

In the process of interpolation in logarithms, values are 
inserted as if the change in the logarithm between the nearest 




40 



FIG. 1. 



tabulated values were directly proportional to the change in the 
number. This assumes that the graph of y = log x for the 
interval concerned is a straight line. From the graph it is 
apparent this would be approximately true. In other words, 
when a number is changed by an amount that is very small 
in comparison with the number itself, the change in the value of 
the logarithm of the number is very nearly proportional to the 
change in the number. 

EXERCISES 

1. Plot the graph of y = logs x. 



Hint, logs XN = 



logio x 
logic 5' 



2. Plot the graph of x 

3. Plot the graph of x 



logs y. 
logs y. 



110. MISCELLANEOUS EXERCISES 

Find by use of logarithms the results of the following exercises. In 
each case make a complete outline or form before using the tables. 



110] 



MISCELLANEOUS EXERCISES 



239 



1. 3.87 X 57.6. 

2. 7.0928 X 0.0052683. 

3. 22.9 X 4.95 X 0.643. 

4. 0.0063982 X 23.473 X 0.062547. 
76.9 



6. 



6. 



(41.911)* 



3.14 



_ 
0.8236' 



17. 



^"(37215)' X 0.78356 
(89.1)3- x (0.764) - 2 



0.6634' 



18 (L 9036 ) 1 - 1 



/ (0.50267 3 



8. 



49.36 X 0.7657 
8.439 



6.47 X 12.93 X 0.2462 
9< "896 X 0.0074939 

10. (0.09245) 3 . 

11. v/0.002855. 

12. v / 0.007(K)08. 

13. (0.935) S. 

14. (4.267) 4 . 
16. (19.26) 1 2 . 



(0.00 14 1 23) 9 
19. (-0.091111)-?. 
45.86 X (0.7288)* 



20. 
21. 
22. 
23. 
24. 



(-9.423)5 

(-a49J73)j 
\X- 207799 ' 



) ~ 15 

\/O7285 + ( _ 
318.2 X : (006004)'"" 

(0.8195)- 3 + (0.9713) 4 
(5.004)-* 



' k)g"4727' 

9fi log 0.87189 
log 0.022223" 



27. The radius r of the inscribed circle of a triangle in terms of its 
sides fij b, and c is given by 

a)(s b)(s c) 

8 

where s = y(a + b + c). Compute r when (a) a = 0.525, b = 0.261, 
c = 0.438; (b) a = (598.2, b = 476.3, c = 744.9; (c) -a = 3.0023, 
b = 2.1128, c = 1.5007. 

28. The number n of revolutions per minute of a certain water 
turbine is given by 



T . ol-v A j 

* h l 3p-0.4 

w ~ 61.3 ' 



240 LOGARITHMS [CHAP. XI 

where h is the height of fall in feet, and P is the horsepower developed. 
Compute n when h = 15 ft. and P = 98 hp. 

29. The formula y = 0.0263Z 1 - 1 gives the relation between y and x 
when x stands for the stress in kilograms per square centimeter of cross 
section of a hollow cast-iron tube subject to tensile stress and y for 
the elongation of the tube in terms of -Q^Q cm. as a unit. Compute 
y when x = 101.8. 

30. The formula y = ks'g*', where log k = 5.03370116, log s = 
-0.003296862, log g = -0.00013205, log c = 0.04579609, gives the 
number living at age x in Hunter's Makehamized American Experience 
Table of Mortality. Find, to such a degree of accuracy as you can 
secure with a five-place table of logarithms, the number living (a) at 
age ten, (b) at age thirty. 

31. Given that 1 km. = 0.6214 mile. Find the number of miles 
in 2489 km. 

32. Given that 1 km. = 0.6214 mile and that the area of Illinois is 
56,625 square miles. Express the area of Illinois in square kilometers 
(to four significant figures). 



CHAPTER XII 
THE SLIDE RULE 

111. Introduction. This chapter, while giving a brief review 
of the method of using a slide rule, stresses the settings relating 
to trigonometry. The settings given apply to most slide rules, 
but the explanation is based on the manuals written by the 
authors of this text for the slide rules manufactured by the 
Keuffel and Esser Company. For a logarithmic explanation of 
this slide rule and more detail concerning the settings, the student 
is referred to the manuals just cited. 

Efficient operation of a slide rule is a comparatively simple 
matter. Since nearly every setting is based on one principle 
called the proportion principle, it is easy to recall forgotten 
settings and devise new ones especially suited to the work at 
hand. The first step is to learn to read the scales on the rule. 

112. Reading the scales.* Figure 1 represents, in skeleton 
form, the fundamental scale of the slide rule, namely the D scale. 



D 


1 

1 


1 
2 


1 
3 


1 1 1 
4 56 


1 
7 


1 
8 


9 


1 



FIG. 1. 

An examination of this actual scale on the slide rule will show 
that it is divided into 9 parts by primary marks that are num- 
bered 1, 2, 3, . . . , 9, 1. The space between any two primary 
marks is divided into ten parts by nine secondary marks. These 
are not numbered on the actual scale except between the primary 
marks numbered 1 and 2. Figure 2 shows the secondary marks 
lying between the primary marks of the D scale. On this scale 
each italicized number gives the reading to be associated with 

* The description here given has reference to the 10-in. slide rule. How- 
ever, anyone having a rule of different length will be able to understand his 
rule in the light of the explanation given. 

241 



242 THE SLIDE RULE [CHAP. XII 

its corresponding secondary mark. Thus, the first secondary 
mark after 2 is numbered 21, the second 22, the third 23, etc. ; the 
first secondary mark after 3 is numbered 31, the second 32, etc. 
Between the primary marks numbered 1 and 2 the secondary 
marks are numbered 1, 2, . . . , 9. Evidently the readings 
associated with these marks are 11, 12, 13, . . . , 19. Finally 
between the secondary marks, see Fig. 3, appear smaller or 
tertiary marks that aid in obtaining the third digit of a reading. 
Thus between the secondary marks numbered 22 'and 23 there 
are four tertiary marks. If we think of the end marks as repro- 



I i j T I 1 I I I [ [ ' i" i i 1 1 1 ji ni m 1 1 jinif nil nnifTTii iiif inijiiiiiiiiijiiiijiiiiiiiiiifiiif 
D I i 2 3 4 5 ft 7 8 9 25S33!5S$g3a333 4 3 5 li 6 !S 7 R 8 8 981 



FICJ. 2. 

senting 220 anH 230, the four tertiary marks divide the interval 
into five pa-ts, each representing two units. Hence with these 
marks we associate the numbers 222, 224, 226, and 228; similarly 
the tertiary marks between the secondary marks numbered 
32 and 33 are read 322, 324, 326, and 328, and the tertiary marks 
between the primary marks numbered 3 and the first succeeding 
secondary mark are read 302, 304, 306, and 308. Between any 
pair of secondary marks to the right of the primary mark num- 
bered 4, there is only one tertiary mark. Hence, each smallest 
space represents five units. Thus the primary mark between tho 
secondary marks representing 41 and 42 is road 415, that between 



|.,,.|....j,...|.,.,|,...|ny,.|H.i|_ 
LJ 1 Ij '2 '3 '4 '5 Ig 1-7 'g !g O C*i eo ^c I <o oo 3> ^^^SJ^ 1 t :28224 5 fi 

Scale D 



FIG. 3. 

the secondary marks representing 55 and 56 is read 555, and the 
first tertiary mark to the right of the primary mark numbered 4 
is read 405. The reading of any position between a pair of 
successive tertiary marks must be based on an estimate. Thus a 
position halfway between the tertiary marks associated with 222 
and 224 is read 223, and a position two-fifths of the way from tho 
tertiary mark numbered 415 to the next mark is read 417. The 
principle illustrated by these readings applies in all cases. 

It is important to note that the decimal point has no bearing 
upon the position associated with a number on the C and D scales. 



114] DEFINITIONS 243 

Consequently, the number G in Fig. 4 may he read 207, 2.07, 
0.000207, 20,700, or any other number whose principal digits 
are 2, 0, and 7. The placing of the decimal point will be explained 
later in this chapter. 

For a position between the primary marks numbered 1 and 2, 
four digits should be read; the first three will be exact and the 
last one estimated. No attempt should be made to read more 
than three digits for positions to the right of the primary mark 
numbered 4. 

While making a reading, the learner should have definitely in mind 
the number associated with the smallest space under consideration. 
Thus between 1 and 2, the smallest division is associated with 10 
in the fourth place; between 2 arid 3, the smallest division has a 



II E D G 

II II 


C F B A I 

\ 1 


1" 


'i 'j ! k' 't,' !;' a 1 ly 2 ' ' ' 


jiii,|iii|iiii,ii-.j,,.,:,-,i| i^-nji.^..^^-^^!. IIIYI.I.J.I 

1 .1 45 



I'm. 4. 

value 2 in the third place; while to the right of 4, the smallest 
division has a value 5 in the third place. 

The learner should read from Fig. 4 the numbers associated 
with the marks lettered .4, B, C, . . . and compare his readings 
with the following numbers: A 365, B 327, C 263, D 1745, E 1347, 
F 305, G 207, H 1078, / 435, ./ 427. 

113. Accuracy of the slide rule. From the discussion of 112, 
it appears that we read four figures of a result on one part of the 
scale and three figures on the remaining part. This means an 
attainable accuracy of roughly one part in 1000 or one-tenth of 
I per cent. The accuracy is nearly proportional to the length 
of the scale. Hence we associate with the 20-in scale an accu- 
racy of about one part, in 2000, and with the Thaeher Cylindrical 
slide rule, an accuracy of about one part in 1 0,000. The accuracy 
obtainable with the 10-in. slide rule is sufficient for most practical 
purposes; in any case the slide rule result serves as a check. 

114. Definitions. The central sliding part of the rule is called 
the slide, the other part, the body. The glass runner is called the 



244 THE SLIDE RULE [CHAP. XII 

indicatory and the line on the indicator is referred to as the 
hairline. 

The mark associated with the primary number 1 on any scale is 
called the index of the scale. An examination of the D scale 
shows that it has two indices, one at the left end and the other at 
the right end. 

Two positions on different scales are said to be opposite if, 
without moving the slide, the hairline may be brought to cover 
both positions at the same time. 

115. Multiplication. The process of multiplication may be 
performed by using scales C and D. The C scale is on the slide, 
but in other respects it is like the D scale and is read in the same 
manner. 

To multiply 2 by 4, 

to 2 on D set index of C, 
push hairline to 4 on C, 
at the hairline read 8 on D. 

Figure 5 shows the setting in skeleton form. 



Fio. 5. 

To multiply 3X3, 

to 3 on D set index of C, 
push hairline to 3 on C, 
at the hairline read 9 on D. 

See Fig. 6 for the setting in skeleton form. 



9 



FIG. 6. 

To multiply 1.5 X 3.5, disregard the decimal point and 

to 15 on D set index of C, 

push hairline to 35 on C, 

at the hairline read 525 on D. 



116] EITHER INDEX MAY BE USED 245 

By inspection we know that the answer is near 5 and is there- 
fore 5.25. 

283 





C' 


!HM| 


,U,lJ 




I i 


i ( 




1*1 


u 


.1 


UUUi 


III 
3 4 


|1|I T] 


e 


\ i 


I 










1675 




474 







FIG. 7. 

To find the value of 16.75 X 2.83 (see Fig. 7) disregard the 
decimal point and 

to 1675 on D set index of (7, 
push hairline to 283 on C, 
at the hairline read 474 on D. 

To place the decimal point we approximate the answer by not- 
ing that it is near to 3 X 16 = 48. Hence the answer is 47.4. 
These examples illustrate the use of the following rule. 

Rule. To find the product of two numbers: To either number 
on scale D set index of scale C, push hairline to second number on 
scale C, at the hairline read product on scale D. Disregard the 
decimal point while making the settings and readings; finally 
place the decimal point in accordance with the result of a rough 
approximation. 

EXERCISES 

1. 3 X 2. 8. 2.03 X 167.3. 

2. 3.5 X 2. 9. 1.536 X 30.6. 

3. 5 X 2. 10. 0.0756 X 1.093. 

4. 2 X 4.55. 11. 1.047 X 3080. 
6. 4.5 X 1.5. 12. 0.00205 X 408. 

6. 1.75 X 5.5. 13. (3.142) 2 . 

7. 4.33 X 11.5. 14. (1.756) 2 . 

116. Either index may be used. It may happen that a product 
cannot be read when the left index of the C scale is used in the rule 
of 115. This will be due to the fact that the second number of 
the product is on the part of the slide projecting beyond the body. 
In this case reset the slide using the right index of the C scale in 
place of the left, or use the following rule: 



246 THE SLIDE RULE [CHAP. XII 

When a number is to be read on the D scale opposite a number 
on the slide scale and cannot be read, push the hairline to the index 
of the C scale inside the body and draw the other index of the C scale 
under the hairline. The desired reading can then be made. This 
very important rule applies generally. 

If, to find the product of 2 and 6, we set the left index of the 
C scale opposite 2 on the D scale, we cannot read the answer on 
the D scale opposite 6 on the C scale. Hence, we set the right 
index of C opposite 2 on D] opposite 6 on C read the answer, 12, 
on D. 

Again, to find 0.0314 X 564, 

to 314 on D set the right index of C Y , 

push hairline to 564 on C, 

at the hairline read 1771 on /). 

A rough approximation is obtained by finding 0.03 X 600 = 
18. Hence the product is 17.71. 

EXERCISES 

Perform the indicated multiplications. 

1. 3 X 5. 6. 0.0495 X 0.0267. 

2. 3.05 X 5.17. 6. 1.876 X 92(5. 

3. 5.56 X 634. 7. 1.876 X 5.32. 

4. 743 X 0.0567. 8. 42.3 X 31.7. 

117. Division. The process of division is performed by using 
the C and D scales. 



l 



V 



Fiu. 8. 

To divide 8 by 4 (see Fig. 8) 

push hairline to 8 on D, 

draw 4 of C under the hairline, 

opposite index of C read 2 on D. 

To divide 876 by 20.4, 

push hairline to 876 on Z), 
draw 204 of C under the hairline, 
opposite index of C read 429 on D. 



118] USE OF SCALES DF AND CF 247 

The rough calculation 800 -s- 20 = 40 shows that the decimal 
point must ho placed after the 2. Hence the answer is 42.9. 

EXERCISES 

Perform the indicated operations. 

1. 87.5 + 37.7. 6. 2875 -f- 37.1. 

2. 3.75 -s- 0.0227. 7. 871 *- 0.468. 

3. 0.085 -f- 8.93. 8. 0.0385 -5- 0.001402. 

4. 1029 + 9.70. 9. 3.14 * 2.72. 

5. 0.00377 -5- 5.29. 10. 3.42 4- 81.7. 

118. Use of scales DF and CF (folded scales). If your slide 
rule contains folded scales, they may often he used to save using 
the italicized rule of 116 to move the slide its own length left- 
ward or rightward. These folded scales are used precisely like 
the other scales. The following rule will indicate how one may 
transfer operations from the C and D scales to the CF and DF 
scales. 

Rule. Shifting an operation from the C and D scales to the CF 
and DF scales or vice versa may be made whenever the process is 
pushing the hairline to a number, never when a number on the slide 
is to be drawn under the hairline. 

For example, to find 2X6, 

to 2 on D set left index of C, 

push hairline to 6 on CF, 

at the hairline read 12 on DF. 

To find 6.17 X 7.34, 

to 617 on DF set index of CF, 

push hairline to 734 on C, 

at the hairline read 45.3 on D. 

By using the CF and DF scales we saved the trouhle of moving the 
slide as well as the attendant source of error. This saving, enter- 
ing as it does in many ways, is a main reason for using the folded 
scales. 

The folded scales may he used to perform multiplications and 
divisions just as the C and D scales are used. Thus, to f}nd 
6.17 X 7.34, 



248 TUP: SLIDE RULE [CHAP. XII 

to 617 on DF set index of CF, 
push hairline to 734 on CF, 
at the hairline read 45.3 on DF] 
or 

to 617 on DF set index of CF, 

push hairline to 734 on C, 

at the hairline read 45.3 on Z>. 

Again to find the quotient 7.68/8.43, 

push hairline to 768 on DF, 
draw 843 of CF under the hairline, 
opposite the index of CF read 0.912 on DF] 
or 

push hairline to 768 on DF, 

draw 843 of CF under the hairline, 

opposite the index of C read 0.912 on D. 

It now appears that we may perform a multiplication or a divi- 
sion in several ways by using two or more of the scales C, D, CF, 
and DF. The sentence written in italics near the beginning of the 
article sets forth the guiding principle. A convenient method of 
multiplying or dividing a number by TT ( = 3.14 approx.) is 
based on the statement: any number on DF is TT times its opposite 
on D, and any number on D is I/TT times its opposite on DF. 

EXERCISES 

Perform each of the operations indicated in exercises 1 to 11 in four 
ways; first by using the C and D scales only; second by using the CF 
and DF scales only; third by using the C and D scales for the initial 
setting and the CF and DF scales for completing the solution; fourth 
by using the CF and DF scales for the initial setting and the C and D 
scales for completing the solution. 

1. 5.78 X (1.35. 9. 0.0948 ^ 7.23. 

2. 7.84 X 1.065. 10. 149.0 4- 63.3. 

3. 0.00465 4- 73.6. 11. 2.718 * 65.7. 

4. 0.0634 X 53,600. 12. 783?r. 

6. 1.769 -f- 496. 13. 783 4- TT. 

6. 946 + 0.0677. 14. 0.0876rr. 

7. 813 X 1.951. 16. 0.504 4- TT. 

8. 0.00755 -s- 0.338. 16. 1.072 -f- 10.97. 



119] 



THE PROPORTION PRINCIPLE 



249 



119. The proportion principle. The proportion principle is 
very important because settings can be devised and recalled by 
using it. When the slide is set in any position, the ratio of any 
number on the D scale to its opposite on the C scale is the same as 
the ratio of any other number on D to its opposite on C. This is 
true because each ratio, in accordance with the setting for division 
is equal to the number on D opposite the index of C. For exam- 
ple, draw 1 of C opposite 2 on D and find the opposites indicated 
in the following table: 



C (or CF) 


I 


1.5 


2 5 


3 


4 


5 


D (or DF) 


2 


3 


5 


6 


8 


10 



Now consider the proportion 



x 9 
56=*- 



(D 



If 9 on C be set opposite 7 on Z), then x will appear on C opposite 
56 on D. Hence, to find in (1), 

push hairline to 7 on Z), 
draw 9 of C under the hairline, 
push hairline to 56 on D, 
at the hairline read 72 on C, 



or 



push hairline to 9 on D, 
draw 7 of C under the hairline, 
push hairline to 56 on C, 
at the hairline read 72 on D. 

Again consider the continued proportion 

C 3.15 x 57.6 _ 
D' 



5.29 4.35 y 183.4 
Observe that 3.15/5.29 is the known ratio, and 

push hairline to 529 on D, 
draw 315 of C under the hairline; 
opposite 435 on D, read x = 2.69 on C, 
opposite 576 on C, read y = 96.7 on D, 
opposite 1834 on D, read z = 109.2 on C. 



250 THE SLIDE RULE [CHAP. XII 

The positions of the decimal points were determined by noticing 
that each denominator had to be approximately twice its numer- 
ator since 5.29 is approximately twice 3.15. The position of the 
decimal point is always determined by a rough approximation. 

Whenever an answer cannot be read because the slide projects 
beyond the body, use the italicized rules of 116 and 118. 

EXERCISES 

Find, in each of the following equations, the values of the unknowns. 

1 2 = *-. o *-.-- ! 

3 7.83 *' 1.804 " 25 ~~ 0.785* 

x _ 246 _ 28 
3> 709 ~ y " 384* 

_x_ _y __ 5.28 _ 2.01 
*' 0.204 ~ 0.506 ~~ z ~~ 0.1034* 

3 z fi 8 ' 5i 

b " 



-- = - _ - = 

' 2.07 ~ 61.3 ~ 1.571' " 1.5" x ~ y ' 

17 1.365 4.86 x y _ 3.75 

7t x ~ "8.53 y y ~ 7.34 ~ 29.7' 

x _ z_ __ y_ U)76 
8 * 49.6 " y " 3.58 " 6.287' 

120. Use of the CI scale. The scale marked CI is designed so 
that when the hairline is set to a number on the CI scale, its recipro- 
cal (1 divided by the number) is set on the C scale. Accordingly 
this scale may be used to deal with reciprocals. Thus, to find x 
when 

x = 415 X 1.87 X 2.54, 
divide through by 415 and replace 2.54 by 1 -f- (1/2.54) to get 

D t x L87 

C : 415 T/2.54* 

Hence, in accordance with the proportion principle, 

push hairline to 1.87 on D, 

draw 2.54 of CI under the hairline, 

push hairline to 415 on C, 

at the hairline read x = 1970 on D. 



121] COMBINED MULTIPLICATION AND DIVISION 251 

Observe; that 1/2.54 of C was drawn under tho hairline indirectly 
by drawing 2.54 on CI under the hairline. If one keeps in mind 
the italicized statement he will find that he can multiply by the 
reciprocal of a number, divide by it, or use it in a proportion by 
using the CI scale for the number instead of the C scale. The 
same 1 principle governs the use of the GIF scale. 

EXERCISES 

In each of the following equations find the value of the unknown : 

1- & " ""' 6 ' y = (2)(4fl)(82). 






= 3.8. 



a 3.41 

3. // -j2S(TTT). 8 ' ?/ - (1.72)(tt.3l)' 

/ 1 \ Q (- 72 ) 

4. y - 74.5 (^23)' ^ G r ).81)(6.43)" 
6. ?/ - (32 1) (40.2) (4.93). 10. y - (!,)(14)(/ 5 ). 

121. Combined multiplication and division. The importance 
of this article is secondary only to 110, which relates to the 
proportion principle. 

, ^ T- i .1 i r 7.36 X 8.44 
Example 1. Find the value of ^ --- - 

Solution. Reason as follows: first divide 7.36 by 92, and then 
multiply the result by 8.44. This would suggest that we 

push hairline to 736 on />, 
draw 92 of C under the hairline; 
opposite 8.44 on C, read 0.675 on D. 

, o T- ! ,1 i , 18 X 45 X 37 
Example 2. Find the value of OQ - - on 

^o X &\j 

Solution. Reason as follows: (a) divide 18 by 23, (6) multiply 
the result by 45, (c) divide this second result by 29, (d) multiply 
this third result by 37. This argument suggests that we 

push hairline to 18 on Z), 

draw 23 of C under the hairline, 



252 THE SLIDE RULE [CHAP. XII 

push hairline to 45 on C, 

draw 29 of C under the hairline, 

push hairline to 37 on C, 

at the hairline read 449 on D. 

To determine the position of the decimal point write 

20 X 40 X 40 , , KA ,, . . . 

Oft v> Q/V = about 50. Hence the answer is 44.9. 
20 X oil 

A little reflection on the procedure of Example 2 will enable the 
operator to evaluate by the shortest method expressions similar 
to the one just considered. He should observe that: the D scale 
was used only twice, once at the beginning of the process and onco 
at its end; the process for each number of the denominator consisted 
in drawing that number, located on the C scale, under the hairline; 
the process for each 7iumber of the numerator consisted in pushing the 
hairline to that number located on the C scale. 

If at any time the indicator cannot be placed because of the projec- 
tion of the slide, apply the rule of 116, or carry on the operations 
using the folded scales. 

Example 3. Find the value of 1.843 X 92 X 2.45 X 0.584 X 
365. 

Solution. Write the given expression in the form 

1.843 X 2.45 X 365 

"(1/92) (1/0.584) 

and reason as follows: (a) divide 1.843 by (1/92), (6) multiply the 
result by 2.45, (c) divide this second result by (1/0.584), (d) 
multiply the third result by 365. This argument suggests that 
we 

push hairline to 1843 on D, 
draw 92 of CI under the hairline, 
push hairline to 245 on C, 
draw 584 of CI under the hairline, 
push hairline to 365 on C, 
at the hairline read 886 on D. 

To approximate the answer we write 2(90) (5/2) (6/10) 300 = 
81,000. Hence the answer is 88,600. 



122] SQUARE ROOTS 233 

EXERCISES 
1375 X 0.0642 65.7 X 0.835 



76,400 ' 3.58 

45.2 X 11.24 A 362 

336" 3.86 X 9.( 

24.1 



218 10. 

3t 4.23 X 50.8' 261 X 32J 

75.5 X 63.4 X 95 

*' 3^6"xlL54* 3 ' U 

5. 2.84 X 6.52 X 5.19. 12 - 



6. 9.21 X 0.1795 X 0.0672. 13. '- 

7. 37.7 X 4.82 X 830. g.g 2 x 6< 95 X 7.85 X 436 

79.8 X 0.0317 X 870 
15. 187 X 0.00236 X 0.0768 X 1047 X 3.14. 

0.917 X 8.65 X 1076 X 3152 
16 ' 7840 



122. Square roots. The square root of a given number is a 
second number whose square is the given number. Thus the 
square root of 4 is 2, and the square root of 9 is 3, or, using the 
symbol for square root, \/4 = 2, and \/9 = 3. 

The A scale consists of two parts that differ only in slight 
details. We shall refer to the left-hand part as A left and to the 
right-hand part as A right. Similar reference will be made to the 
B scale. 

Rule. To find the square root of a number between 1 and 10, set 
the hairlijie to the number on scale A left and read, its square root at 
the hairline on the D scale. To find the square root of a number 
between 10 and 100, set the hairline to the number on scale A right 
and, read its square root at the hairline on the D scale. In either case 
place the decimal point after the first digit. A similar statement 
relating to the B scale and the C scale holds true. For example, 
set the hairline to 9 on scale A left, read 3 ( = \/9) a ^ the hairline 
on D, set the hairline to 25 on scale B right, read 5 ( = \/25) at 
the hairline on C. 



254 THE SLIDE RULE [CHAP. XII 

To obtain the square root of any number, move the decimal point 
an even number of places to obtain a number between 1 and 1 00 ; then 
apply the rule written above in italics; finally move the decimal point 
one half as many places as it was moved in the original number but 
in the opposite direction. * The learner may also place the decimal 
point in accordance with information derived from a rough 
approximation. 

For example, to find the square root of 23,400, move the 
decimal point four places to the left, thus getting 2.34 (a number 
between 1 and 10); set the hairline to 2.34 on scale A left; read 
1.530 at the hairline on the D scale; finally, move the decimal 
point ^ of 4 or two places to the right to obtain the answer 
153.0. The decimal point could have boon placed after observ- 
ing that \/10,006 = 100 or that \AO,000 = 200. Also, the 
left B scale and the C scale could have been used instead of the 
left A scale and the D scale. 

To find \/3850, move the decimal point two places to the left 
to obtain \/38.50; set the hairline to 38.50 on scale A right; read 
6.20 at the hairline on the D scale; move the decimal point one 
place to the right to obtain the answer 62.0. The decimal point 
could have been placed by observing that \/3600 = 60. 

To find V^O. 000585, move the decimal point four places to the 
right to obtain \/5-85; find \/5785 = 2.42; move the decimal 
point two places to the left to obtain the answer 0.0242. 

EXERCISES 

1. Find the square root of each of the following numbers: 8, 12, 17, 
89, 8.90, 890, 0.89, 7280, 0.0635, 0.0000635, 63,500, 100,000. 

2. Find the length of the side of a square whose area is (a) 53,500 ft. 2 ; 

(b) 0.0776 ft. 2 ; (c) 3.27 X 10 7 ft. 2 

3. Find the diameter of a circle having area (a) 256 ft. 2 ; (b) 0.773 ft. 2 ; 

(c) 1950 ft. 2 

123. Combined operations involving square roots. When the 
hairline is set to a numbei* on the B scale it is automatically set 
on the C scale to the square root of the number. Therefore the 

* The following rule may also be used: If the square root of a number 
greater than unity is desired, use A left when it contains an odd number of 
digits to the left of the decimal point; otherwise use A right. For a number 
less than unity use A left if the number of zeros immediately following the 
decimal point is odd; otherwise, use A right. 



123] COMBINED OPERATIONS 255 

B scale can be used in combined operations like the CI scale. 
Naturally, the rule for square-root settings should be used to 
determine whether B left or B right is to be used in any particular 
case. The following example will illustrate the method of 
procedure. 

.,,.,. V832 X \/365 X 1863 
Example. Evaluate 



X 89,400 
Solution. In accordance with italicized statement of 121, 

push hairline to 832 on A left, 
draw 736 of CI under the hairline, 
push hairline- to 365 on B left, 
draw 894 of C under the hairline, 
push hairline to 1863 on CF, 
at the hairline read 8460 on DF. 

The method of finding cube roots is much like that of finding 
square roots. The following rule may be used: 


Rule. To obtain the cube root of a number, move the decimal 
point over three places (or digits) at a time until a number between 1 
and 1000 is obtained. Then push the hairline to the new number 
on K left, K middle, or K right according as it lies between 1 and 10, 
10 and 100, or 100 and 1000. Read the cube root on scale D at the 
hairline and place the decimal point after the first digit. Then 
move the decimal point one-third as many places as it was moved 
in the original number but in the opposite direction. 

EXERCISES 



2.38 *' 0.275 XT 

86 X \/734 X TT (2.60) 2 

775 X V0^85 ' 2.17 X 7.28 

20.6 X 7.89 2 X 6.79 2 
4.67 2 X 281 

189.7 X V6.00296 X A/347 X 0.274 
X 165 X TT 



7. A/285 X 667 X \/6.65 X 78.4 X V0.00449. 
239 X \/O677 X 374 X 9.45 X IT 

O. " * 



84.3 X V9350 X V28400 



256 THE SLIDE RULE [CHAP. XII 

124. The S (sine) and ST (sine tangent) scales. The numbers 
on the sine scales S and ST* represent angles. In order to set the 
indicator to an angle on the sine scales it is necessary to determine 
the value of the angles represented by the subdivisions. Thus, 
since there are six primary intervals between 4 and 5, each 
represents 10'; since each of the primary intervals is subdivided 
into five secondary intervals, each of the latter represents 2'. 
Again, since there are five primary intervals between 20 and 25, 
each represents 1; since each primary interval here is subdivided 
into two secondary intervals, each of the latter represents 30'; 
since each secondary interval is subdivided into three parts, these 
smallest intervals represent 10'. These illustrations indicate the 
manner in which the learner should analyze the part of the scale 
involved to find the value of the smallest interval to be con- 



ST 



0fi8' 1W 26' 450* 



i ou ia 

k-4 



SS 

1820' 



& 



!! 



C 



o o 



FIG. 9. 

sidered. In general, when setting the hairline to an angle, tho 
student should always have in mind the value of the smallest 
interval on the part of the slide rule under consideration. 

When the indicator is set to a black number (angle) on scale S or 
STj the sine of the angle is on scale C at the hairline and hence on 
scale D when the indices on scales C and D coincide. 

When scale C is used to read sines of angles on ST, the left index 
of C is taken as 0.01, the right index as 0.1. In reading sines 
of angles on S, the left index of C is taken as 0.1, the right index as 
1. Thus, to find sin 3626', opposite 3626' on scale S, read 
0.594 on scale C; to find sin 324', opposite 324' on scale ST, 
read 0.0593 on scale C. Figure 9 shows scales STj S, and C on 
which certain angles and their sines are indicated. As an exer- 
cise, read from your slide rule the sines of the angles shown in the 
figure and compare your results with those given. 

* The ST scale is a sine scale, but since it is also used as a tangent scale 
it is designated ST. 



125] THE T (TANGENT) SCALE 257 

EXERCISES 

1. By examination of the slide rule verify that on the S scale from the 
left index to 16 the smallest subdivision represents 5'; from 16 to 30 
it represents 10'; from 30 to 60 it represents 30'; from 60 to 80 
it represents 1; and from 80 to 90 it represents 5. 

2. Find the sine of each of the following angles: 

(a) 30. (c) 320'. (e) 8745'. (g) 1438'. (i) ll48'. 

(b) 38. (d) 90. (/) 135'. (h) 2225'. (j) 5130'. 

3. Find the cosine of each of the angles in Exercise 2 by using the 
relation cos <p = sin (90 <p). 

4. For each of the following values of x, 

(a) 0.5, (c) 0.375, (e) 0.015, (g) 0.062, (i) 0.92, 

(6) 0.875, (d) 0.1, (/) 0.62, (h) 0.031, (j) 0.885, 

find the value of <p less than 90, (A) if <p = sin" 1 x, where sin" 1 x means 
"the angle whose sine is x"; (B) if <p = cos" 1 x. 

6. Find the cosecant of each of the angles in Exercise 2 by using the 

relation esc <p = g^- 

Hint. Set the angle on S, read the cosecant on CI (or on DI when 
the rule is closed). 

6. Find the secant of each of the angles in Exercise 2 by using the 

relation sec <p = 

7. For each of the following values of x, 

(a) 2. (b) 2.4. (c) 1.7. (d) 6.12. (e) 80.2. (/) 4.72. 
find the value of <p less than 90, (A) if <p = esc" 1 x; (B) if <p = sec" 1 x. 

125. The T (tangent) scale. When the indicator is set to a 
black angle on scale T, the tangent of the angle is on scale C at the 
hairline and hence on scale D when the indices of scales T and 
D coincide. Also when the indicator is set to a black angle on 
scale T, the cotangent of the angle is on scale CI at the hairline. 
Thus, to find tan 36, push the hairline to 36 on T] at the hair- 
line read 0.727 on C. To find cot 2710', push the hairline to 
2710' on T} at the hairline read 1.949 on CI. 

When scale C is used to read tangents, the left index is taken 
as 0.1 and the right index as 1.0. Only those angles that range 



258 



THE SLIDE RULE 



[CHAP. XII 



from 543' to 45 appear on scale T. It is shown in trigonometry 
that for angles less than 5 43', the sine and tangent are approxi- 
mately equal. Hence, so far as the slide rule is concerned, the 
tangent of an angle less than 543' may he replaced by the sine 
of the angle. Thus to find tan 215', push the hairline to 215' 
on ST, at the hairline read 0.0393 on C. To find the tangent 
of an angle greater than 45, use the relation 

cot 6 = tan (90 - 6). 

To find tan 56, push the hairline to 34 ( = 90 - 56) on T, at 
the hairline read 1.483 on CI. The student should observe that 
he could have set the hairline to 56 in red on the T scale and 
thus have avoided subtracting 34 from 90. 

EXERCISES 
1. Fill out the following table: 



<f> 


86' 


2715' 


t>219' 


17' 


741,V 


87 


4728' 


tan <p 
















cot <p 

















2. The following numbers are tangents of angles. Find the angles. 



(a) 0.24. 
(6) 0.785. 
(c) 0.92. 



(d) 0.54. 

(e) 0.059. 
(/) 0.082. 



(g) 0.432. 
(h) 0.043. 
(i) 0.0149. 



0') 0.374. 
(k) 3.72. 
(/) 4.67. 



(ro) 17.01. 
(n) 1.03. 
(o) 1.232. 



3. The numbers in Kxercise 2 are cotangents of angles, 
angles. 



Find the 



126. Combined operations. The method for evaluating 
expressions involving combined operations as stated in 121 and 
123 applies without change when some of the numbers are 
trigonometric functions. This is illustrated in the following 
example. 



Example. Evaluate 



6.1VT7 sin 72 tan 20 
2.2 



127] SOLVING A TRIANGLE 

Solution. Write 

\/J7 sin 72 tan 20 

2.2 ro 



259 



Push hairline to 17 on A right, 

draw 2.2 of C under the hairline, 

push hairline to 20 on T, 

draw 6.1 of CI under the hairline, 

push hairline to 72 on S, 

at the hairline read 3.96 on D. 



EXERCISES 



Evaluate the following: 
18.6 sin 36 



" sin 21 

2 32 sin 18 
27.5 

3 4.2 tan 38 
sin 453()' " 

34.3 sin 17 
tan2230'' 

13.1 cos 40 




tan3510' 
fi 17.2 cos 35 


cot 50 
- 7.8csc3530 / 


cot 2125' 
8 63.1 sec 80 
* " tan 55 ' 
g sin 18 tan 20 
* 3.7 tan 41 sin 31 
sin2625' 



13. 
14. 



(K61_c8c 12_1 
cot3516' 
1 sin 2240 / 
tan 28T6 r " 



16 3.1 sin 6135' esc 1518_' 
cos'"27"40' cot 20 >~~' 



sn 



tan 3010' 
0.0037 sin 4950 / 
tan 26 / 

A/16.5 sin 4530' 



.2 cot 71*10' 



8.1 tan 22 18' 

11. 3.14 sin 1310' esc 32. 

12. 7. ITT sin 4735'. 



16. 
17. 
18. 

19. 
20. 

21. 

(19.1)(7.61)\/69.4 

22. (48.1)(1.68) sin 39. 

23. 0.0121 sin 81 cot 41. 
1.01 cos 7110' sin 15 



\X(U 4.91 

tan 1314'' 

sin 5130' 



(39.1)(6.28) 
esc 4930' 



24. 



-V/4.81 cos 2710' 



127. Solving a triangle by means of the law of sines. If the 

sides and angles of a triangle are lettered as indicated in Fig. 10, 



260 



THE SLIDE RULE 



[CHAP. XII 



the law of sines is written 

sin A _ sin B _ sin C 

a ~~ b c 

This law is the basis of most slide-rule solutions of triangles. 

C 



(2) 




To solve the triangle shown in Fig. 11 for a and c, write 

sin 65 sin 48 _ sin 67 

258 " a "" c ' 

and, using the setting based on the proportion principle, 

push hairline to 258 on D, 

draw 65 of 8 under the hairline, 

push hairline to 48 on S, 

at the hairline read a = 212 on D, 

push hairline to 67 on S, 

at the hairline read c = 262 on D. 

The decimal point was placed by inspection. 

In general, to solve any triangle in which a side and the angle 
opposite are known, 

push hairline to known side on D, 
draw opposite angle of S under the hairline, 
push hairline to any known side on D, 
at the hairline read opposite angle on S, 
push hairline to any known angle on S, 
at the hairline read opposite side on D. 

When an angle A of a triangle is greater than 90, replace it by 
180 A. This is permissible since sin (180 A) = sin A. 
When the decimal point in a result cannot be placed by inspec- 
tion, compute the part involved approximately by using (2) with 
the trigonometric functions replaced by their natural values. 



127] SOLVING A TRIANGLE 261 

When the given parts of a triangle are two sides and the angle 
opposite one of them, there may be two solutions. For example, 
if the given parts are a = 175, b = 215, A = 3530', the two 




3530 



t 



-c 2 -J I 
Cl J 

FIG. 12. 

possible triangles are shown in Fig. 12. Using the setting (2) 
of 127, 

push hairline to 175 on D, 

draw 3530' of S under the hairline, 

push hairlino to 215 on D, 

at the hairline read BI = 4530' on S. 

Compute Ci = 180 - 3530' - 4530' = 99 

push hairline to 81 (= 180 - 99) on S, 

at the hairline read c\ = 298 on D, 

Compute C 2 = B l - 3530' = 10, 

push hairline to 10 on S, 

at the hairline read c 2 = 62.3 on D. 

EXERCISES 

Solve the following oblique triangles. 



1. a = 50, 
A = 65, 
B = 40. 


5. a = 120, 
b = 80, 
A = 60. 


9. 6 = 91.1, 
c = 77, 
B = 517'. 


2. c = 60, 
A = 50, 
B = 75. 


6. b = 0.234, 
c = 0.198, 
B = 109. 


10. a = 50, 
c = 66, 
A = 123 !!'. 


3. a = 550, 
A = 1012', 
B = 4636'. 


7. a = 795, 
A = 7959', 
B = 4441'. 


11. a = 8.66, 
c = 10, 
A = 5957'. 


4. a = 222, 
b = 4570, 
C = 90. 


8. a = 21, 
A = 410', 
B = 75. 


12. 6 = 8, 
a =120, 
A - 60. 



262 



THE SLIDE RULE 



[CHAP. XII 



13. A ship at point S can be seen from each of two points, A and B, 
on the shore. If AB = 800 ft., angle SAB = 6743', and angle SKA 
742r, find the distance of the ship from A. 

14. To determine the distance of an inaccessible tower A from a point 
B, a line BC and the angles ABC and BOA were measured and found 
to be 1000 yd., 44, and 70, respectively. Find the distance A K. 

Solve the following oblique triangles. 



15. 



16. 



a = 18, 
b = 20, 
A = 5524'. 
b = 19, 
c = 18, 
C = 1549'. 



17. 



18. 



a = 32.2, 

c = 27.1, 

C = 5224'. 

b = 5.16, 

r = 6.84, 

/* = 443'. 



19. 



20. 



a 177, 
6 = 216, 
.1 = 353(>'. 
a = 17,060, 
b - 14,050, 
B = 40. 




Fio. 



21. Find the length of the perpendic- 
ular p for the triangle of Fig. 13. 
How many solutions will there be for 
triangle ABC if (a) b = 3? (6) 6 = 4? 

(c) b = p? 



128. To solve a right triangle when two legs are given. When 
the two legs of a right triangle are the given parts, first find the 
smaller acute angle from its tangent, and then apply the law of 
sines to complete the solution. 

Example. Solve the right triangle of Fig. 1 4 in which a = 3.18, 
B b = 4.24. 

Solution. From the triangle 
^3.18> 



6=4.24 

Fia. 14. 



read tan A 



a =3.18 



) and write 



this equation in the form 

tan A 1 
"3.T8 



4.24 



Using the setting based on the principle of proportion, 

set index of C to 4.24 on D, 

push hairline to 3.18 on ), 

at the hairline read A = 3652 / on T. 

Since angle A = 3652' and a = 3.18, we know a pair of opposite 
parts and may proceed to use the law of sines. Since the hairline 



129] TO SOLVE A TRIANGLE 263 

is on 3.18 of D from the setting just made, 

draw 3652' of 8 under the hairline, 
at index of C read c = 6.31 on D. 
Evidently B = 90 - A = 638'. 

The following rule states the method of solution. 

Rule. To solve a right triangle for which two legs are given, 

set index of C to larger leg on D, 

push hairline to smaller leg on D, 

at the hairline read the smaller acute angle on T, 

draw this angle on S under the hairline, 

at index of slide read hypotenuse on D. 

EXERCISES 

Solve the following right triangles: 

1. a = 12.3, 4. a = 273, 7. a = 13.2, 
b = 20.2. b = 418. b = 13.2. 

2. a = 101, 6. a = 28, 8. a = 42, 
b = 116. b = 34. b = 71. 

3. a = 50, 6. a = 12, 9. a = 0.31, 
6 = 23.3. 6 = 5. b = 4.8. 

129. To solve a triangle in which two sides and the included 
angle are given. The method here explained will consist in 
dividing the given triangle into two right triangles by means of 
an altitude to one of the known sides and then solving the two 
right triangles separately. The method is illustrated in the 
following example. 

Example. Solve the triangle of Fig. 15 in which a = 6.18, 
6 = 9.27, C = 32. 




--6 = 9.27 

FIG. 15. 



264 THE SLIDE RULE [CHAP. XII 

Solution. Draw the altitude BD to side AC, and observe that 
angle BCD = 90 and a = 6.18 are known. Hence use the 
italicized rule of 127 and 

set index of C to 6.18 on D, 

push hairline to 32 on S f 

at the hairline read p = 3.27 on Z), 

opposite 58 ( = 90 - 32) on S read n = 5.24 on D, 

compute m = 9.27 - 5.24 = 4.03. 

To solve triangle ABD, use the italicized rule of 128. Hence 

set index of C to 4.03 on D, 

push hairline to 3.27 on D, 

at the hairline read A = 393' on 7 7 , 

draw 393' on S under the hairline, 

at index of C read c 6.19 on D. 

Evidently B = 180 - 32 - 393' = 10857'. 

If the given angle is obtuse the altitude lies outside the triangle, 
but the method is essentially the same as that used in the solution 
above. 

EXERCISES 

Solve the following triangles 

l.o = 94, 4. b = 2.30, 7. a = 0.085, 

b = 56, c = 3.57, c = 0.0042, 

C = 29. A = 62. B = 5630'. 

2. a = 100, 6. a = 27, 8. a = 17, 
c = 130, c = 15, 6 = 12, 

B = 5149'. B = 46. C = 5918'. 

3. a = 235, 6. a = 6.75, 9. b = 2580, 
6 = 185, c = 1.04, c = 5290, 

C = 8436'. /* = 1279'. A = 13821'. 

10. The two diagonals of a parallelogram are 10 and 12 and they 
form an angle of 49 18'. Find the length of each side. 

11. Two ships start from the same point at the same instant. One 
sails due north at the rate of 10.44 miles per hour, and the other due 
northeast at the rate of 7.71 miles per hour. How far apart are they 
at the end of 40 min.? 

130. To solve a triangle in which three sides are given. 
When three sides of a triangle are given, one angle may be found 



131] TO CHANGE RADIANS TO DEGREES 265 

by using the law of cosines, 

a 2 = b 2 + c 2 2bc cos A, 

and the other parts may then be found by means of the law of 
sines. 

Example. Solve the triangle of Fig. 16 in which a = 15, 
b = 18, c = 20. 

Solution. From the law of cosines we 

write . ft , % 

b**Vy \a=15 

A J>2 | 9 2 

\~ = We = 




___ _ 

2 X 18 X 20 720 FIG. 10. 

Hence, using a setting based on the proportion principle, 

to 720 on D set 499 of C, 

at index of D read A = 466' on S (red). 

Now complete the solution by means of the law of sines to obtain 
B = 5964', C = 74. When all three angles are read from the 
slide rule, the relation A + B + C = 180 may be used as a 
check. Thus, for the solution just completed, 

A + B + C = 466' + 5954' + 74 = 180. 

EXERCISES 

Solve the following triangles : 

1. a = 3.41, 3. a = 35, 6. a = 97.9, 
6 = 2.60, 6 = 38, 6 = 106, 
c = 1.58. c = 41. c = 139. 

2. a = 111, 4. a = 61.0, 6. a = 57.9, 
6 = 145, 6 = 49.2, b = 50.1, 
c = 40. c = 80.5. c = 35.0. 

131. To change radians to degrees or degrees to radians. 

Since TT (= 3.1416 approx.) radians equal 180, we may write 

IT _ r (number of radians) 
180 ~ d (number of degrees) 



266 THE SLIDE RULE [CHAP. XII 

Hence 

draw IT on C opposite 180 on Z), 
push hairline to d (number of degrees given) on D, 
at the hairline read number of radians on C, 
push hairline to r (number of radians given) on C, 
at the hairline read number of degrees on D. 

EXERCISES 

1. P^xpress the following angles in radians: 

(a) 45. (d) 180. (0) 223()'. 

(6) 00. (f) 120. (h) 200. 

(r) 90. (/) 135. (i) 3000. 

2. Express the following angles in degrees: 

(a) 7T/3 radians. (r) TT '72 radian. (?) 20ir/3 radians. 

(6) .'W4 radians. (</) TTT (> radians. (/) (UMr radians. 

3. Kxpress in radians the following angles: 

(a) 1. (r) 1". (?) 18034'20". 

(6) r. (d) ioir. (/) :K)2r) f 4;r'. 

4. Find the following angles in degrees and minutes: 

(a) TIT radian; (b) 2-^ radians; (c) 1.0 radians; (d) radians. 



Sl'HEliK 1 AL TRIGONOMETRY 



CHAPTER XIII 
THE RIGHT SPHERICAL TRIANGLE 

132. Introduction. Just as plane trigonometry has for its 
object the study of the relations existing among the sides and 
angles of a plane triangle, so spherical trigonometry has for its 




Chart your course right 
(.Courtesy, John Hancock Mutual Life Insurance Company) 

object the study of the relations connecting the sides and angles 
of a spherical triangle. Since the earth is approximately a 
sphere, this theory will apply when distances and directions on the 
earth are in question. Hence the subject of spherical trigonom- 
etry is basic in navigation, geodesy, and astronomy. 

133. The spherical triangle. The circle in which a plane 
through the center of a sphere intersects the sphere is called a 

269 



270 THE RIGHT SPHERICAL TRIANGLE [CHAP. XI II 

great circle. As in plane geometry, an arc on a great circle is 
measured by the angle that it subtends at the center and will be 
expressed in degrees, minutes, and seconds. 

A spherical triangle consists of three arcs of great circles that 
form the boundaries of a portion of a spherical surface. As in 
plane geometry, the vertices of the spherical triangle will be 
denoted by capital letters A, B, and C and the sides opposite by 
a, b, and c, respectively. The magnitude of an angle of a spher- 
ical triangle is that of the plane angle formed by tangents to the 
sides of the angle at its vertex. In general, we shall consider only 
spherical triangles, each of whose sides and each of whose angles is 
less than 180. 

The planes of the great circles belonging to a spherical triangle 
form a trihedral angle at the center of the sphere (see Fig. 1). 

The face angles of this trihedral angle, 
being measured by their intercepted 
arcs, are designated by the same let- 
ters as the corresponding sides of the 
spherical triangle. The tangents to 
the arcs AB and AC at point A, being 
perpendicular to the radius OA, are 
the sides of the plane angle of dihedral 
angle M-AO-N. These tangents 
measure angle A of the spherical tri- 
angle ABC. Hence an angle of the 
spherical triangle is measured by the dihedral angles made by the 
planes of its sides. 

Important propositions from solid geometry : 

1. The sum of the angles of a spherical triangle is greater than 
180 and less than 540; that is, 180 < A + B + C < 540. 

2. // two angles of a spherical triangle are equal, the sides opposite 
are equal; and conversely. 

3. // two angles of a spherical triangle are unequal, the sides 
opposite are unequal, and the greater side lies opposite the greater 
angle; and conversely. 

4. The sum of two sides of a spherical triangle is greater than the 
third side. 

EXERCISES 

1. If each angle of a spherical triangle is a right angle, what is the 
value of each side? 




FORMULAS OF THE RIGHT SPHERICAL TRIANGLE 271 

2. Show that if a spherical triangle has two right angles, the sides 
opposite these angles are quadrants and the third angle has the same 
measure as the opposite side. 

3. The face angles of the trihedral angle associated with a spherical 
triangle are each 90 and the radius of the sphere is 10 in. Find the 
angles of the triangle in degrees, and find the sides both in degrees and 
in inches. 

4. Find the magnitude of the face angles and of the dihedral angles 
of the trihedral angle associated with a spherical triangle whose sides 
are 90, 90, and 00. 

6. The face angles of a trihedral angle at the center of the earth are 
50, 6()38', 4550'20". Find in nautical miles* the lengths of the sides 
of the associated spherical triangle on the surface of the earth. 

6. Two great circles on a sphere intersect at an angle of 2330'. 
Find the least great-circle distance from the pole of one to a point on 
the other. 

7. What can be said regarding the size and shape of a spherical equi- 
angular triangle if the sum of its angles is (a) nearly equal to 180; 
(b) nearly equal to 540? 

8. Find all sides and angles of a spherical triangle having as angles 
A = 90 q , B = 90, and 

( ) C = 30. (c) C = 120. (e) C = 110. 

(b) C = 45. (d) C = 70. (/) C = 160. 

9. Show that the sum of the angles of a right spherical triangle is 
greater than 180 and less than 360. 

134. Formulas relating to the right spherical triangle. Since 
spherical triangles having more than one right angle can be solved 
by inspection, we shall be concerned with right spherical triangles 
having only one right angle. 

In this article, ten formulas relating to the right spherical 
triangle are derived, and in the next article simple rules for writing 
these formulas are given. 

The solution of all cases of spherical triangles generally con- 
sidered in spherical trigonometry can be solved by means of 
these formulas. 

In Fig. 2 is represented a spherical pyramid that is part of 
a sphere having unit radius and center 0. In the right spherical 
triangle ABC bounding the base of the pyramid, C is a right angle, 

* A nautical mile is the length of an arc of a great circle on a sphere the 
size of the earth subtended by an angle of 1' at its center. 



272 



THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 



and therefore the dihedral angle having edge OC is a right dihedral 
angle. From A, a plane is passed perpendicular to edge OB 
cutting the spherical pyramid in the triangle AED. Since 
OE is perpendicular to plane AED, it is perpendicular to lines 
EA and ED. Hence angle A ED is the plane angle of the dihedral 
angle having OB as edge. Therefore angle AED is equal to 
angle B. Also, plane AED is perpendicular to plane COB, since 
it is perpendicular to a line in the plane. Therefore line AD is 




FIQ. 2. 

perpendicular to plane OBC because it is the intersection of the 
two planes OAD and ADE, both of which are perpendicular to 
OBC. Hence the angles ADO and ADE are right angles. Each 
face angle of the trihedral angle 0-ABC is measured by the side 
of the spherical triangle intercepted by it and is therefore desig- 
nated by the same letter as that side. 
From Fig. 2 we read 

= cos ft. (I) 



DA . . EA OE 

= sin 6, -j- = sin c, = cos c, 



1 



Also from triangle OED, ED/OE = tan a. Replacing OE in 
this by cos c from (I) and simplifying slightly, we have 



ED = OE tan a = cos c tan a. 
Similarly, from triangle OED, 

ED = OD sin a = cos 6 sin a. 



(ID 



CUD 



Figure 3 is obtained from Fig. 2 by enlarging it and writing on it 
the values of the line segments just derived. 



134] FORMULAS OF THE RIGHT SPHERICAL TRIANGLE 273 



Both values for ED, one from (II) and the other from (III) are 
written on ED. From the triangle A ED in Fig. 3, we read 



. n sin b 
sin B = 



n 
cos B 



sin c 

tan a cos c 

- -. - j 
sin c 

, D sin b 

tan = -.- - =7 
sin a cos o 

tan o cos c = sin a cos 6. 



(IV) 




FIG. 3. 

These last four equations may be written in the following form: 

sin b = sin c sin B, (1) 

cos B = tan a cot c, (2) 

sin a = tan b cot B, (3) 

cos c = cos a cos b. (4) 

Similarly, by passing a plane through B of Fig. 2 perpendicular 
to OA and proceeding as above, we could prove the formulas 



sin a = sin c sin 4, 
cos A = tan b cot c, 
sin b = tan a cot 4. 



(5) 
(6) 
(7) 



Formulas (5), (6), and (7) are the result of interchanging a and b 



274 



THE RIGHT SPHERICAL TRIANGLE [('HAP. XIII 



and A and B in (1), (2), and (3), respectively. From (7) cot A = 
sin 6/tan a and from (3) cot B = sin a/tan 6; multiplying these 
two equations member by member, we obtain 

^ A r> sin b ^, sin a . 

cot A cot B = 7 X T r = cos b cos a. 

tan a tan b ' 

or, interchanging members and replacing cos b cos a by cos c 
from (4), 

cos c = cot A cot 5. (8) 



Similarly from (2), (5), and (4), we obtain 
cos B = cos b sin A 

and from (6), (1), and (4), 

cos A = cos a sin B. 



(9) 
(10) 




136. Napier's rules. The ten formulas derived in 134 need 
not be memorized, for it is easy to write them by using two rules 
devised by John Napier, the inventor of loga*- 
rithms. 

Figure 4 represents a right spherical triangle. 
Figure 5 contains the same letters as Fig. 4 
except C(= 90), arranged in the same* order. 
The bars on the letters c, B, and A mean the 
complement of; thus B means 90 B. Note 
that the barred parts arc the hijpotcnmc and 
the two angles each cf which has a side along 
The angular quantities a, 6, c, A, B are called 
There are two circular parts contiguous with 
any given part and two parts that are not con- 
tiguous to it. Speaking of this given part as 
the middle part, we call the two contiguous 
parts the adjacent parts, and the two non-con- 
tiguous parts the opposite parts. Napier's 

rules may now be stated as follows: 

r 

Napier's Rule I. The sine of any middle 
part is equal to the product of the cosines of the opposite parts. 
Napier's Rule II. The sine of any middle part is equal to the 
product of the tangents of the adjacent parts. 



the hypotenuse, 
the circular parts. 




135J NAPIER'S RULES 275 

We may use the expression sin middle cos opposite = 
tan adjacent as an aid in recalling these rules. 

Thinking of any part as the middle part, we can write two 
formulas, one from each of the two rules. Considering each of 
(lie five parts in turn as middle part, we may write ten formulas, 
and these are found to be the ten formulas numbered (1) to (10) 
iu 134.* 

Example. Use Napier's rules to write two formulas by using 
(a) b as middle part; (b) A as middle part. 

Solution. Noting that sin A = sin (90 A) = cos A, 
cos A = cos (90 A) = sin A, etc., and applying the first rule 
to the parts bj c, B (see Fig. 6), 
\\c obtain 

sin b = cos (7 cos B, 
or 

sin b = sin c sin B. (a) 

Applying the second rule, us- 
ing parts A, 6, , we obtain 

sin 6 = tan A tan a = cot A tan a. (b) 

Similuily, using the parts A, B, a and the first rule, and after- 
wards the parts c, A, b and the second rule, we obtain 

sin A = cos B cos n, or cos A = sin B cos a, (c) 
sin A = tan c tan 6, or cos A = cot c tan b. (d) 

The formulas (a), (6), (c), and (d) are, respectively, the formulas 
(1), (7), (10), and (G) of 134. 

EXERCISES 

1. Solve each of the following right spherical triangles for the 
unknown part indicated. 

( a ) a = 30, 
b - 60, 

(b) c = GO , 
a = 45, 

(c) a = 45, 
/? - (10, 

<: After the student has become familiar with the use of Napier's rules, 
lie may x,w him- hv writing the desired formulas directly from the triangle 
on which the letters have been properly barred. 








(d) 


a 


= GO , 






c = 


? 




B 


= 30, 


A = 


? 






w 


c 


= GO , 






B = 


? 




A 


= 45, 


b = 


? 






(/) 


A 


= 30, 






c = 


? 




B 


= 60, 


a = 


? 



276 



THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 




2. Using Fig. 7, show that for- 
B mulas (1) to (10) hold true for the 
case a is greater than 90, c is greater 
than 90, 6 is less than 90. 



3. Solve each of the following right spherical triangles for the 
unknown part indicated: 



(a) a = 60, 
b = 120, 

(b) c = 135, 
b = 120, 

(c) B = 150, 

c = 120, 



A = ? 



(d) A = 135, 
B = 60, 

(e) a = 30, 
B = 120, 

(/) c = 120, 

a = 135, 



c = ? 
A = ? 
B = ? 



4. Corresponding to each of the following formulas pertaining to a 
plane right triangle, write, using Napier's rules, an analogous formula 
pertaining to a right spherical triangle. 



(a) sin A = a/c. 

(b) sin B = b/c. 

(c) 1 = cot A cot B. 



(d) cos A = b/c. 

(e) cos B = a/c. 



(/) tan A = a/b. 
(g) tan B = b/a. 



6. On Fig. 8 interchange A and B, also a and 6. Then express the 
values of the line segments OD, OE, BE, BD, DE in terms of a, 6, c, 




Fia. 8. 



and write each of these line values on the figure. Equate two values of 
DE to obtain formula (4), and apply the definitions of the trigonometric 
functions to triangle BDE to obtain formulas (5), (6), and (7). 



136] TWO IMPORTANT RULES 277 

6. Using formula (4), show that the hypotenuse of a right spherical 
triangle is less than or greater than 90, according as the two legs lie 
in the same quadrant or in different quadrants. 

7. Using formula (10), show that in a right spherical triangle each 
leg and the opposite angle are of the same quadrant. 

8. Use Napier 's rules to write a formula involving the following, 
taking c as unknown part, 

(a) c, B, A. (b) c, B, a. (c) c, B, b. 

9. Use Napier's rules to write three formulas, each involving 
a and b. 

10. Prove that tan A = 



11. Prove that cos A 



tan b cos c 
sin 6 cos a 
sin c 



136. Two important rules. In what follows it will be con- 
venient to speak of an angle of the first quadrant or of the second 
quadrant. An angle is said to be of the first, second, third, or 
fourth quadrant according as its terminal side falls in the first, 
second, third, or fourth quadrant when laid off in the usual 
manner relative to rectangular coordinate axes. 

From formula (10) of 134, namely, 

cos A = cos a sin B, 

it follows that cos A and cos a must have the same sign since 
sin B is positive in all cases. Hence both A and a must be less 
than 90, or both must be greater than 90. Formula (9) may 
be used to show that B and b must be of the same quadrant. The 
following rule expresses the relation. 

Rule (A). In a right spherical triangle an oblique angle and 
the side opposite are of the same quadrant. 

From formula (4), namely, 

cos c = cos a cos Z>, 

it appears that the product cos a cos 6 must be positive when c is 
less than 90; therefore cos a and cos b must have the same 
sign, and for that reason a and b are both of the first quadrant 
or both of the second quadrant. From the same formula it 
appears that cos a cos b must be negative when c is greater than 



278 



THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 



90; therefore cos a and cos b must have opposite signs, and a and 
b are of different quadrants. The following rule expresses the 
relation. 

Rule (B). When the hypotenuse of a right spherical triangle 
is less than 90, the two legs are of the same quadrant; when the 
hypotenuse is greater than 90, one leg is of the first quadrant 
and the other of the second. 

Rules (.1) and (B) enable the computer to tell the quadrant of 
an angle found from its sine or its cosecant. 

EXERCISES 



State the quadrant of each of the unknown parts in each of the 
right spherical triangles indicated in the following diagram: 





a 


b 


c 


A 


H 


1 


30 


40 








2 


30 




120 






3 


120 








r>o" 


4 




140 


75 






5 








120 


130" 


6 




35 




100 




7 






100 


100 




8 






60 




60 



137. Solution of right spherical triangles. When two pnrts 
of a right spherical triangle in addition to the right angle are 
given, the remaining parts can be computed from formulas 
obtained by using Xapier's rules. In solving the triangle it will 
be found advantageous to proceed as follows: 

a. Draw a right spherical triangle lettered in the conventional 
way and encircle the given parts. 

fe. Write a formula for each unknown part by applying Napier's 
rules. Each formula should contain the unknown part and both 



137] SOLUTION OF RIGHT SPHERICAL TRIANGLES 



270 



of the given parts. Then write a check formula connecting the 
three required parts. 

c. Make a form. 

d. Fill in the blank spaces of the form. 

Example. Solve the right spherical triangle in which a 
6659'31", b = 15634'19". 





Fus. 10. 



Solution. Figures 9 and 10 display the circular parts of a 
right spherical triangle, the known parts a, b being encircled. 
Using Napior's rules, in connection with Fig. 10, wo write 



sin = tan cot A, 
sin = tan cot B, 
cos c = cos cos , 
cos c = cot A cot B. 



or cot A = sin opt @, 
or cot B = sin cot , 



(a) 

(&) 

(c) 
(d) 



The symbols / sin, I cot, etc., written in any line of a form mean 
log sine of the angle at the left of the line, log cotangent of that 
angle, etc. For convenience the negative part 10 of the 
characteristic will bo omitted in the forms. 

Tho symbol ( ) written before a logarithm in any form calls 
attention to tho fact that the antilogarithm of that logarithm 
is negative. Hence an odd number of symbols ( ) appearing 
in a column used to evaluate a product by logarithms will 
indicate that the product is negative*. An even number of 
symbols ( ) will indicate a positive product. 

In the forms of spherical trigonometry we shall omit the expres- 
sions a =, 6 =, etc., to save space. The student will understand 
that each symbol refers to the number at the extreme left of its line. 

The computation of the unknown parts from the formulas (a), 
(6), (c), and the check by (d) is displayed on page 280. 



280 



THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 



a - 6659'31" 
b = 15634'19" 


(a) 
Zcot 9.62802 
I sin 9.59944 


(W 
2 sin 9.96400 
Zcot (-)O. 36319 


(c) 
icos 9.59202 
/cos (-)9. 96264 


A 8025'01" 


/cot 9.22746 






B - 15447'26" 


Jcot (-)0. 32719 


/cot (-)0. 32719 




c ~ 1111'0" 


Zcos (-)9. 55465 




/cos (-)9. 55466 



Observe that the check obtained by adding log cot A to 
log cot B to get log cos c checks only the logarithms of the 
computed parts. Errors made in finding A, B, and c from associ- 
ated logarithms would not affect the check. 

EXERCISES 

Solve the following right spherical triangles: 



B 


= 1032', 
= 123'. 


2. c 

B 


= 4640', 
= 2050'. 


3. a 

B 


= 11854', 
= 1219'. 


4. a 

c 


= 4327', 
= 6024'. 


5. b 
c 


= 4836 / , 
= 6942'. 


6. a 

c 


= IGS^S^S", 
= ISO^^O". 


7. c 
# 


= 11248', 


8. c 


- 3234 r , 



11. 



12. 



a 
b 

13. a 
B 

14. 

15. 



A = 4444'. 
9. ^L = 1163r25 ; ', 
B = 11643'12". 
= 5454 / 42 // , 

c = 



10. 



559'32", 
22157". 
3627', 
4332 / 31 // . 
2946 / 8", 
13724'21". 
a = 14427 / 3", 
6 = 328 / 56 /f . 
b = 3627', 
a = 4332'31 // . 

16. A = 6315'12", 
B = 13533'39". 

17. A = 6754 / 47 / ', 
B = 9957'35". 

18. b = 2215'7", 
c = 559 / 32 // . 

19. a - HS^O'lO", 
5 = 9536'. 

20. 6 = 9247 / 32", 
A = 



21. If angle A of a right spherical triangle is 28, what is the maximum 
size of angle J5? 

22. A ship leaves point M\ in Fig. 11 sailing due 
c east and follows a great-circle track to a point M 2 . 

If Mi is in latitude 4030 / N., longitude 75 W. and if 
M 2 is in longitude 60 W., find the distance D 
traveled, the latitude of M 2, and the course angle C 
at M 2 . 

Hint. The angle DLo at the north pole P n is the 
Fio. 11. difference in the longitudes of the two points Mi 




138] 



THE AMBIGUOUS CASE 



281 



and AT 2 . The distances from the points Mi and M 2 to P n are respec- 
tively the complements of the latitudes of these points. 

C 

23. In the spherical triangle ABC (Fig. 

12), p is the arc of a great circle perpen- 
dicular to side c. Write an expression for 
B in terms of A, a, and b. 

Hint. 
them. 



Find two values of p and equate 




FIG. 12. 



24. If in the triangle ABC of Exercise 23, A = 4010', a = 
and 6 = 6450', find B. 

C 



25. All lines in Fig. 13 represent arcs of 
great circles. Find all unknown parts, thus 
solving a spherical triangle for which two 
angles and the included side are given. 



4620', 





Fio. 14. 



138. The ambiguous case. When the given parts are a side 
and the angle opposite, two solutions are obtained. In such 
cases each unknown part is found 
from the sine and hence may be 
chosen either in the first quad- A< 
rant or in the second quadrant; 
that is, in the case of each un- 
known an angle and its supple- 
ment must be written. If A and a represent the given parts and 
C the right angle, the two triangles will form a lune as indicated in 
Fig. 14; for in this figure B f appears as 180 - B, c 1 as 180 - c, 
and 6' as 180 - b. 

The solution of the following example will illustrate the 
method of finding a double solution when it exists. 

Example. Solve the right spherical triangle in which 
a - 4645', A - 5912'. 

Solution. Using Napier's rules in connection with Fig. 15 we 
obtain 



282 



THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 




sin c = sin esc , 
sin B = sec cos Q, 
sin 6 = tan cot , 
sin 6 = sin c sin #. Check 



(a) 
(b) 

(0 



The solution is displayed below. 



a = 4645' 
A = 5912' 
ci = 6759 / 30 // 
c 2 = 1220 30" 
B! = 4821'27" 
B 2 = 13138'33" 
61 = 3919 / 24 // 
6 2 = UO^O'SG' 7 


(a) and (check) 
1 Z sin 9.86235 
] Z esc 0.06603 


(b) 
t sec 0.16419 
I cos 9.70931 


(c) 
Z tan 0.02655 
Z cot 9.77533 


/ sin 9.92838 
I sin 9.87350 


I sin 9.87350 


/ sin 9.80188 


Z sin 9.801 88 



The six answers were grouped to obtain the solutions 61, GI, BI, and 
6 2 , C 2 , 2 by using the rules (A) and (J3) of 136. 

EXERCISES 
Solve the following right spherical triangles: 

1. b = 3544 ; , 4. a = 7721'50 // , 
5 = 3728 ; . A = 8356'40 ;/ . 

2. 6 = 129 33 r , 6. a = 160, 
B = 10459'. A = 150. 

3. 6 = 2139 ; , 6. 6 = 4218 / 45 // , 
J5 = 4210 / 10 // . B = 4615 / 25' / . 

7. Apply Napier's rules to Fig. 15 to obtain a formula involving the 
known parts a, A, and the unknown part 6. From this formula show 
that there may be no solution. Discuss the case that arises when 
a and A are supplementary. 

Solve the following right spherical triangles : 

8. 6 = 4218 / , 9. a = 2010', 
B = 4218'. A = 11520'. 

139. Polar triangles. The poles of a great circle on a sphere 
are the points where a perpendicular to the plane of the great 



139] 



POLAR TRIANGLES 



283 



circle through its center pierces the surface of the sphere. To 
obtain the polar triangle of a spherical triangle ARC, construct 
three great circles on the sphere having their poles at A, B. and 
C. Two arcs, one having B as pole and the other C as pole, 
intersect in two points on opposite sides of arc BC. Denote by 
A' 



C' 





A' the point that lies on the same side of the great circle through 
BC as A. Locate B f and C' by an analogous procedure. Then 
triangle A'B'C' is the polar of triangle ARC. Figures 16 (a) and 
16 (6) indicate the relations. 

The following theorems from solid geometry are important: 

1. // A'B'C' represents the polar triangle of spherical triangle 
ABC, then ABC is the polar triangle of A'B'C'. 

2. An angle of any spherical triangle is the supplement of the 
opposite side in the polar triajigle. 

In accordance with Theorem 2, we have the following relations 
between the sides and angles represented in Figs. 16 (a) and (b) : 



A' = 180 - a, 
B' = 180 - 6, 
C' = 180 - c, 



A = 180 - a', 
B = 180 - 6', 
C = 180 - c'. 



(11) 



If in an equation containing the quantities a, 6, c, A, B, C, 
these quantities be replaced by their values in terms of a', 6', 
c! , A 1 , B', C', from (11), a new equation having reference to the 
polar triangle is obtained. The relations (11) will be used in the 
next article to solve a spherical triangle having a side equal to 90. 

EXERCISES 

1. Use relations (11) to find the parts of the polar triangle of each of 
the following spherical triangles. 



284 THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 

(a) A = 13559.1', B = 10010.1', C = 9843.3', c = 90, a = 

13520', b = 9831.5'. 
(6) a = 5416.0', b = 11447.0', C = 7035.9', c = 90, A = 

4957.9', B = 1215.5'. 

(c) a = 11635.6', 6 = 10514.8', c = 4317.2', A = 11247.4 / , 
^ = 846.7', C = 4459.1'. 

(d) a = ISG^.G', 6 = 4318.5', c = 11443.3', A = 13215.3', 
B = 4719.5', C = 76 48.4'. 

2. For each of the following formulas, write a new formula having 
reference to the polar triangle: 

(a) sin a = sin c sin A. 

(b) tan b = tan c cos A. 

(c) tan a = sin 6 tan A. 

(d) cos c = cos b cos a. 

(e) sin 6 = sin c sin 5. 

(/) cos a = cos 6 cos c + sin 6 sin c cos A. 

(0) cos A = cos B cos C + sin B sin C cos a. 

cos ^(A + B) _ tan ^c 

cos ^(A 5) tan ^(a + b) 

sin ^(A + B) _ tan ^c 

sin ^(A J5) tan ^ (a b) 



3. For each of the following triangles find the known parts of the 
polar triangle; solve these polar triangles: 

(a) c = 90, a = 12248.2', B = 2135.4 ; . 
(6) c = 90, a = 4930.0', B = 6536.2'. 

140. Quadrantal triangles. A spherical triangle having a side 
equal to 90 is called a quadrantal triangle. Evidently the polar 
triangle of a quadrantal triangle is a right spherical triangle. 
Hence this polar triangle can be solved in the usual way, and the 
unknown parts of the quadrantal triangle can then be obtained 
by using relations (11). 

Example. Solve the spherical triangle in which c = 90, 
A = 11538', b = 13958'. 

Solution. Using (11) of 139 we obtain for the polar triangle 
C' = 180 - c = 90, a 1 = 180 - A = 6422', B' = 180 - 
b = 402'. The solution of the polar triangle follows: 



141] 



MISCELLANEOUS EXERCISES 



285 



a' = .6422' 
R f = 402' 


I cot 9.68109 
I cos 9.88404 


Zsin 9.95500 
I tan 9.92433 


I cos 9.63610 
Z sin 9.80837 


c' = 6949'37" 


Z cot 9.56513 






V = 378'25" 


I tan 9.87933 


Z tan 9.87933 




A 1 ** 7350'34" 


I cos 9.44446 




I cos 9.44447 



Using equations (11) again, we obtain C = 180 c f = 
11010'23", B = 180 - V = 14251'36", a - 180 - A' = 
1069'26". 

EXERCISES 

Solve the following right spherical triangles and then use (11) to 
obtain the solution of the polar triangle of each : 

l.o = 1156', 2. a = 11243'30", 

6 = 12314'. c = 8510'10". 

Solve the following quadrantal triangles: 

3. B = 11754 ; 30", 5. A = 15316', 
a = 9542 / 20 // , b = 193', 

c = 90. c = 90. 

4. B = 69 45', 6. b = 159 33'40", 
A = 9440', a = 9518'20", 

c = 90. c = 90. 

B 



7. In Fig. 17 a = 1812', B = 7445', 
c = 90. Solve the right triangle ACD, 
and from it deduce the 'solution of the 
quadrantal triangle ABC. 




Fio. 17. 



141. MISCELLANEOUS EXERCISES 
1. Solve the following spherical triangles: 



(a) a = 3748'12", 

b = 5944'16", 

C = 90. 
(6) A = 11047'50", 

B = 13535 / 34 // , 

c = 90. 



(c) A = 5532 / 45", 
B = 10147'56 /; , 
C = 90. 

(d) b = 13225', 
B = 10730', 
C = 90. 



286 



THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 



(e) B = 7445', 
a = 1812', 
c = 90. 



(/) a = 2518'45", 
A = 1558'15", 
C = 90. 



2. Solve the following isosceles spherical triangles : 



(a) c = 518', (6) 

A = B = 4157'. 



50 19'40", 

5 = iooi2'30". 



Hint. Draw the arc of a great circle through the vertex perpendicular 
to the opposite side. This perpendicular bisects the base and the angle 
at the vertex. 

3. Two great circles on a sphere intersect at 35. A point A on one 
circle is 65 from their intersection. Find the distance from the inter- 
section to the point nearest to A on the other circle. 



4. All lines in Fig. 18 represent arcs 
of great circles. Find all unknown parts, 
thus solving a spherical triangle for which 
two sides and the included angle are given. 

6. All lines in Fig. 19 represent 
arcs of great circles. Find all unknown 
parts, thus solving a spherical triangle 
for which two sides and an angle 
opposite one of them are given. 




6=96 
FIG. 18. 




In Exercises 6 to 15 the terms lati- 
tude and longitude will be used ex- 
tensively. The student should refer to the definitions of these quan- 
tities in 162. 

6. Figure 20 represents a spherical tri- 
angle, with the North Pole at P, Panama 
in latitude 857' N. at Mi, and Honolulu 
in latitude 2l18' N. at M z . M 2 D is the 
arc of a great circle perpendicular to PMi 
and DLo is 7820'. Solve the right triangle 
I completely and afterward triangle II. 
From the results find the distance M iM% 
and the course angle at MI. 





141] MISCELLANEOUS EXERCISES 287 



7. The northern vertex V (see Fig. 21), or 
point of highest latitude reached on the great- 
circle track from M\ to Af 2 , is in latitude L v = 
6827' N., and longitude \ = 2023' W. A ship 
sails on the great-circle track M\M^ starting from 
Mi in longitude Xi - 3718' W. to AT, in longitude 
X 2 = 2628' W. Find the distance M^I*. 

Hint. DLoi = Xi X y , DLoz X 2 X,,' and 
F is a right angle. 

FIG. 21. 

8. (a) If the difference of longitude of two places A and B on the 
earth is 50 and their latitudes are 30, find the distance AB measured 
on the equal latitude circle. 

(6) What is the distance AB measured on a great circle? The 
radius of the earth is approximately 3900 land miles. 

9. Two points A and B are the ends of a 500-land-mile arc of a 
small circle in latitude 30 N. Find the difference in their longitudes. 
If A i and B\ are both in latitude 3(i N. and the arc of a great circle 
connecting them is o()() land miles long, what is the difference in their 
longitudes? Assume the radius of the earth is 3W50 land miles. 

10. The initial course of a certain ship sailing from New York (lati- 
tude L = 4040' X., long. X = 7358'30" W~.) is due east. After 
she has sailed (500 nautical miles on a great circle, find her latitude, 
longitude, and course. 

11. Find the latitude and distance from New York of the ship in 
Kxercise 10 when her longitude is 1,525' W. 

12. Find the latitude and longitude of the northernmost point on a 
great circle track sailed by a ship leaving San Francisco, (latitude 
L = 3828' N., long. X = 12323' W.) on a course of 310. 

13. What is the shortest distance from New York to the great circle 
that passes through San Francisco and the nearest point to San Fran- 
cisco on the 180 meridian? 

14. Find the point on the 180 meridian that is nearest San Francisco 
(latitude L - 3828' N., long. X = 12323' W.)? 

15. A ship sails from a place in longitude 3314'25" W. 2000 nautical 
miles on a great circle. If the initial course is due cast and if the 
change in longitude is 5314'25", find the latitude of departure and the 
course of arrival. 

16. In the case of a right spherical triangle, show that the following 
relations hold true: 



288 THE RIGHT SPHERICAL TRIANGLE [CHAP. XIII 

(a) sin (c 6) sin (c + b) = cos 2 B sin 2 c. 

(b) sin a cos 6 = cos c tan a = sin 6 cot B = sin c cos B. 

(c) cos 2 A + cos 2 B + sin 2 a sin 2 B = 1. 

(d) 2 sin c cos 6 = sin (c + b) sec 2 ?A. 

(e) 2 sin c cos 6 = sin (c b) esc 2 ^A. 
(/) cos A + cos B = sin (a + b) esc c. 

(0) cos B cos A = sin (a 6) esc c. 

(h) cos 5 sin (c + b) sec 2 ^A = tan a cot c sin (c b) esc 2 ?A. 

(1) sin (a + 6) sin c sin A = sin 2 a cos 6 + sin a cos a sin b. 
(j) sec c sec 2A(2 sec 2 A) = sec a sec 6 sec 2 A. 

(k) tan 2 a = tan (c + 6) tan (c - 6) 



CHAPTER XIV 
THE OBLIQUE SPHERICAL TRIANGLE 

142. Law of sines. To prepare for solving spherical triangles, 
we shall develop general formulas analogous to those developed 
in Chaps VII and VIII for plane triangles. 

The law of sines for spherical triangles, analogous to the law 
of sines for plane triangles, may be stated as follows: 

The sines of the sides of a spherical triangle are proportional to 
the sines of the angles opposite , or in symbols 

sin a^ _ sin b _ sin c .^ 

sin A ~~ sin B ~~ sin C 

In Fig. 1 let a, 6, c represent the sides of a spherical triangle 
and let A, B, C represent the opposite angles. Draw an arc 





FIG. 1. 



CD(= h) of a great circle through the vertex C perpendicular to 
the side c, or the side c produced, to form the right spherical 
triangles ACD and BCD. Apply Napier's rules to these right 
triangles to obtain 

sin h = sin 6 sin A, sin h = sin a sin B. 
Equating these two values of sin h, we get 

sin a sin B = sin 6 sin A, 
289 



290 



TI1K OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 



or, dividing by sin A sin B, 



sin a 
sin A 



sin 6 
sin B 



(2) 



In like manner, by drawing an arc, from A perpendicular to CB 
and arguing as above, we can show that 

sin b 



_ sin c 
sin B sin C 



(3) 



Equations (2) and (3) are together equivalent to (1). The law 
of sines may be used in the solution of a spherical triangle 1 when 
a side and the angle opposite are included among the given parts. 

When a part of a spherical triangle is found by moans of the law 
of sines, there is often some difficulty in determining whether the 
part found is of the first quadrant or of the second quadrant; for 
sin A = sin (180 A). Other formulas must be used in many 
cases. However, the following theorems from solid geometry 
will often enable the computer 1o determine the quadrant. 

The order of magnitude of the sides of a spherical triangle is 
the same as the order of magnitude of the respective opposite 
angles; or, in symbols, if 



a < b < c, 



then 



A < B < C. 



The sum of two sides of a spherical triangle is greater than the 
third side. 

EXERCISES 

1. Figure 2 represents the spherical triangle ABC with its associated 

trihedral angle O, the face angles of 
which are a, 6, c. AF is the inter- 
section of two planes, one 
perpendicular to OB. the other 
perpendicular to OC. Point /<' is 
in plane OCR. Taking OA = [ 
unit, express the values of all 
straight-line segments of the figure 
iu terms of a, 6, c, J5, and C. 
Derive the law of sines from the 
result. 

2. Check the following data by using the law of sines: 




FIG. 2. 



(a) A 
b 



10840', B = 13420 ; , C 
15445', c = 349'. 



7()18', a = 14536', 



143] THE LAW OF COSINES FOR SIDES 291 

(6) A = 4721', B = 2220', C = 14640', a = 1!79 / , 6 = 2722', 

c = 13820'. 
(c) A = 11010',# = 13318',C = 70lG',a = 1476', b = 1555', 

c = 3259'. 

3. Use the law of sines to find the missing parts of the following right 
spherical triangles : 

(a) a = 588'19", 6 = 3249'22", B = 3712'53", c - G340'. 

(b) a = 3G14'6", A = 4929'5G", b = 3845', c = 51!'ll". 

4. Use the law of sines to find the missing part of each of the following 
spherical triangles: 

(a) A = 1305'22", B = 3226'G", C = 3645'26", c = 516'12", 

a = 8414'29". 
(6) A = 70, C = 0448'12 // , c = 116, a = 575G / 53 // , 

b = 13720'33 // . 

6. Solve the polar triangles of the triangles of Exercise 3. 

143. The law of cosines for sides. The cosine of any side of a 
spherical triangle is equal to the product cf the cosines of the two 
other sides increased by the product of the sines of the two other sides 
and the cosine of the angle included between them, or in symbols 

cos a = cos b cos c + sin b sin c cos A. (4) 

The following proof is analogous to the one given for the law 
of cosines in plane trigonometry. 

In Fig. 1 let arc AD = <?. Then arc BD = c - <p. Write 
these values on the triangle of Fig. 
1 (a) , and place bars over a, 6, A , and C 

B in preparation for using Napier's 
rules. The result is Fig. 3. 

Now apply Napier's rules to tri- 
angles ACD and BCD to obtain "/ 




cos a = cos h cos (c <p), (5) 

cos b = cos h cos <p. (6) _ 

A- 
Divide (5) by (6) member by mem- FI. 3. 

ber, and transform slightly to get 

cos a _ cos h cos (c <p) __ cos c cos <p + sin c sin ^ 
cos b cos h cos v cos <p 



(7) 



292 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

or, simplifying further, 

cos a = cos 6 (cos c + sin c tan <p). (8) 

Again apply Napier 's rules, using parts b, A, <p of triangle A CD 
to obtain 

cos A = cot b tan ^, 
or 

tan <p = cos A tan &. (9) 

Replace tan <p in (8) by its value from (9) to get 

cos a = cos 6 (cos c + sin c cos A tan 6), (10) 

or, simplifying the right-hand member, 

cos a = cos b cos c + sin b sin c cos 2!. (11) 

Similarly, we may obtain 

cos b = cos a cos c + sin a sin c cos 5, (12) 

cos c = cos a cos b + sin a sin 6 cos (7. (13) 

An argument differing slightly from the one just used shows that 
(11) holds for a triangle shaped like the triangle of Fig. 1(6). 

The law of cosines applies to the solution of a spherical triangle 
when two sides and the included angle are given. Although 
it is not adapted to logarithmic computation, it is used in the 
derivation of many important formulas of spherical trigonometry. 

Example. Find c in the spherical triangle for which a = 
7624'40", 6 = 5818'36", C = 11630'28". 
Solution. The law of cosines may be written 

cos c = cos a cos b + sin a sin 6 cos C. 

Here it will be necessary to compute each product in the right- 
hand member, add the results, and then find c from a table of 
natural cosines; or find the logarithm of the natural cosine, and 
then find c from the table giving the logarithms of cosines. The 
computation is indicated in the following form: 



144] THE LAW OF COSINES FOR ANGLES 293 



o = 7624'40" 
6 = 5818'36" 
C = 11630'28" 



(cos a cos b) (sin a sin b cos C) 



I cos 9.37098 
I cos 9.72042 



I sin 9.98767 
I sin 9.92988 
Zcos(-)9.64965 



0.12342 log 9.09140 

-0.36915 log (-)9.56720 

-0.24573 /. c = cos- 1 (-0.24573) = 10413'30". 

144. The law of cosines for angles. Applying (1 1) to the polar 
triangle (see 139) of ABC, we obtain 

cos a' = cos V cos c 1 + sin V sin c' cos A'. (14) 

Using equation (11) of 139 to replace a', b', c', and A' of (14) by 
180 - A, 180 - B, 180 - C, and 180 - a, respectively, we 
obtain 

cos (180 - A) = cos (180 - B) cos (180 - C) 

+ sin (180 - B) sin (180 - C) cos (180 - a), 
or 

cos A = cos B cos C sin B sin C cos a, 
or 

cos A = cos 5 cos C + sin B sin C cos a. (15) 

Similarly, we obtain from (12) and (13) 

cos B cos A cos C + sin ^4 sin C cos 6, (16) 

cos C = cos A cos # + sin A sin J5 cos c. (17) 

Evidently this process of applying known formulas to the 
polar triangle of a given one is very important. It furnishes a 
method of deriving from every equation applying to a general 
spherical triangle another equation that may be called the dual 
of the first one. The role played by the sides in the given equa- 
tion is played by the angles in the dual equation, and the role 
played by the angles in the given equation is played by the sides 
in the other. A similar statement applies to theorems relating to 
a spherical triangle. This principle of duality will come to our 
attention again and again in the discussion that follows. 

Example. In a certain spherical triangle, A = 60, B = 60, 
and c = 60. Find C. 



294 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

Solution. Substituting 60 for each of the letters A , J9, and c 
in (17), we obtain 

cos C = - cos 60 cos 60 + sin 60 sin 60 cos 60 

--i + t-i- 
Hence 

C - cos" 1 i = 8249'9". 

EXERCISES 

1. Use the law of cosines to find a for each of the following spherical 
triangles : 

(a) b = 60, (b) 6 = 45, (c) 6 = 45, 

c = 30, c = 30, c = 60, 

A - 45. A = 120. A = 150. 

2. Use the law of cosines for angles to find A for each of the follow- 
ing triangles : 

(a) B = 120, (b) B = 135, 

C = 150, C = 120, 

a = 135. a = 30. 

3. In a spherical triangle, given a = 30, b 45, c = 60, find A. 

4. Derive the law of sines algebraically from the law of cosines. 
Hint. Solve (11) for cos A, form sin' 2 A, and reduce the numerator 

to a form involving cosines only. Then show that sin 2 A/sin' 2 a is 
symmetrical in a, b, c. 

5. In Fig. 4, ABC represents a spherical triangle with its associated 

trihedral angle 0. BLM is a plane 
through B perpendicular to OB, 
intersecting OA produced, in M and 
OC produced, in L. Taking OB = 1 
unit, express the values of the line 
segments OL, OA/, BL, BM in terms 
of a, b, c, then apply the law of 
cosines of plane trigonometry to the 
triangles BLM, arid OLM, and equate 
FJG 4 two values of LA/ 2 to obtain after slight 

transformation 

cos b = cos a cos c + sin a sin c cos B. 




146] THE HALF-ANGLE FORMULAS 295 

6. From formula (15) show that 

hav (180 - A) = hav (li + C) - sin Jl sin C hav , 

remembering that hav A = ^0 cos A). 

7. In each of the triangles of Exercise 1 complete the solution by 
means of the law of sines. 

8. Solve the polar triangles of the triangles of Exercises 1 and 3. 

9. Using the law of cosines, prove that in a spherical triangle having 
three sides of the second quadrant the angldp opposite are of the second 
quadrant. 

10. What equations are dual to those expressing the law of sines? 

11. Find the equation dual to the one written in Exercise (). 

12. Replace C' by 90 in (1), (13), (15), and (17), and then obtain 
the resulting formulas by applying Napier's rules to the parts of a 
right spherical triangle. 

145. The six cases. When three parts of a spherical triangle 
are given, the other three parts can be computed. Accordingly 
a classification of spherical triangles is made on the basis of given 
parts. Six cases are referred to as follows: 

I. Given the three sides. 
II. Given the three angles. 

III. Given two sides and the included angle. 

IV. Given two angles and the included side. 

V. Given two sides and an angle opposite one of them. 
VI. Given two angles and a side opposite one of them. 

For purposes of solution, there are, in a sense, only three cases. 
If a method of solution for Case I is known, this same method may 
be applied to solve the polar of a triangle classified under Case II. 
The solution of a quadrantal triangle in 140 by the method of 
solving a right spherical triangle illustrates the process. Simi- 
larly, the formulas used to solve a triangle classified under 
Case III may be used to solve the polar of a triangle classified 
under Case IV; also, the same formulas may be used to solve a 
triangle coming under Case V and the polar of a triangle classi- 
fied under Case VI. 



146. The half -angle formulas. This article is devoted to the 
derivation of formulas that may be used to solve triangles for 



296 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

which the given parts are three sides or three angles. Solving 
(11) for cos A, we have 

, cos a cos 6 cos c /10 x 

cos A = : j : (lo; 

sin o sin c 

Equating 1 minus the left-hand member to 1 minus the right- 
hand member and simplifying slightly, we get 

A sin b sin c + cos b cos c cos a 

1 COS A = : f 9 

sin 6 sin c 

or, replacing sin b sin c + cos b cos c by cos (b c), 

cos (6 c) cos a 



1 cos A = 



sin 6 sin c 



Now, replacing 1 cos A by 2 sin 2 ^A and changing the right- 
hand member by using (36) of 57 and the fact that sin ( 6) = 
sin 0, we get 



(19) 



sin o sin c 
Denote half the sum of the sides by s and write 

8 = i(a + 6 + c). (20) 

Subtracting in succession a, 6, and c from both members of (20), 
we obtain 

s - a = K-a + 6 + c), 5 - b = J(a - b + c),) . 

, - c = i(a + b - c). / (2l) 

Substituting from (21) in (19) and taking the square root of both 
members, we obtain 



sin & = /a* (' -7 ) ** ( - *). (22 ) 

2 \ sin b sin c 



Considerations of symmetry show that 



= / 

\ 



sn - 



sin o sm c 
sn si" 



/si" ( ~ o) si" (jES. (24) 

\ sin o sm & 



146] THE HALF-ANGLE FORMULAS 297 

Similarly, proceeding as above, we obtain 

- . A - . cos a cos b cos c 

I + cos A = 1 H -- . r~ . -- ? 
sin o sin c 

__ cos a (cos b cos c sin b sin c) 

sin 6 sin c 

__ cos a cos (b + c) 
sin 6 sin c 

- + b + C) . (25) 



sin 6 sin c 

Replacing in (25) 1 + cos A by 2 cos 2 \A, using (20) and (21) and 
extracting tin- square root of both members, we get 



sin b sin c 

Considerations of symmetry show that 



(26) 



COS 



cos 



= /sinssinQ,-^ (2?) 

\ sin a sin c 

= J sin ' Sin (S " C) - (28) 

\ sm a sm b 



Dividing (22) by (26), member by member, and replacing 
sin \A -T- cos ^A by tan ^A, we obtain 



tan p = s ~ -g; n - s - - (29) 

^ \ sin s sin (s a) 

Multiplying numerator and denominator under the radical by 
sin (s a) and removing I/sin 2 (s a) from the radical, we have 



tan 4A = I /ffl (* a) an (8 --T) ain (^ c) (3Q) 

^ sm (s a) \ sin s 

or 



where 



/sin ( 

= <* / 

\ 



(s a) sin (s b) sm (s c) 

----- ; -- 

sm s 



298 



THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 



Similarly, 



tan ^5 = 



tan^C 



(33) 
(34) 



sin (s b) 
r 

sin (s c)* 

Since hav A = sin 2 -J4, formula (22) may bo written 

hav A = sin (5 b) sin (5 c) esc b esc c. (35) 

Similar formulas for hav B and hav C may bo obtained from 
(23) and (24). Formula (35) is often used whon haversine tables 
are available. 



147. Cases I and II. Given three sides or given three angles. 

Evidently formulas (31), (33), and (34) are adapted to solve a 
spherical triangle when three sides are given. To solve a spher- 
ical triangle when the three angles are given, we find the sides 
of the polar triangle by subtracting each of the given angles 
from 180 and then applying equations (31), (33), and (34) to 
find the angles of the polar triangle; subtraction of each of these 
angles from 180 gives the sides of the original tiitmgle. Also, 
the formulas of Exercise 1 on page 289 may be used. 

Example. Find A, B, and C for a spherical triangle in which 
a = 7014'20", b = 4924'10", c = 3846'10". 

Solution, s =$(a + b + c) = 7912'20". The solution by 
means of formulas (32), (31), (33), and (34) and the check by 
the law of sines follows. The number in parenthesis above each 
column refers to the formula associated with the column. 

(34) 
I csc 18803 



log 9 35441 



I tan 9 54244 



(32) (31) (33) 


- o = 858'00" 


/ sin 9 19273 


I esc 80727 


s - b = 2948'10" 


I bin 9 69637 




I csc 30363 


s - c = 4026'10" 


I sin 9 81197 






s = 7912'20" 


I csc 00775 






2)log8~70882 






r log 9 35441 


log 9 35441 


log 9 35441 


M - 5525'38", I tan 16168 




A - 11051'16" 





i B => 2428'2", 
B - 4866'4" 
\C - 1913'23" 
C - S826'46" 
Check. 



I /sin 9.97364 

I I sin 9 97058 

00306 



Z BUI 9 88042 

I am 9 . 87735 

(T00307 



I tan 9 65H04 



c ' I sin 9 79671 

clj^unO 79364 

00307 



147] CASES I AND II 299 

EXERCISES 

1. Write o = - + -^-+_~, and use equations (11) of 139 to derive 

g , _ ! + _+, __ 2 . 00 _ A+B+-C = 27()0 _ fi 

s' - n' - 1)0 - (a - A), s' - b' = 90 - (a - B\ 

s' - c' = 90 - (a - ('). 

Then apply equations (22), (26), and (2 ( J) to the polar triangle to 
obtain 



(cos (a B) cos (<r C) 

' " - 



an a = 



A / costr cos (0- 

= \ 



cos <r cos (o- A) 
tan - -~- 



' COS ((7 B) COS ((7 C) 

2. Solve the following spherical triangles: 



(a) = 30, 


(r) a - 150, 


(e) A = 60, 


6 = 45, 


6 = 120, 


B = 30, 


c - 00, 


c = 60. 


C = 120. 


(6) a - 30, 


(rf) A = 60, 


(/) A = 150, 


b = f>(), 


B - 135, 


J5 = 120, 


c = (>0. 


C = (50. 


C = 135. 



3. Solve tlie following spherical triangles: 

(a) a - 110, (e) A = 80, 

b = 32, B = 1 10, 

c = 96. C = 130. 

(6) a = 10814', (/) A = 5955'10", 

fc - 7529 / , B = 8536 / 50 / ', 

c = 5037'. C = SO^S'lO". 

(c) a - 7815'12 /f , (flf) A = 895'46 // , 

6 = 10120 ; 18", B = 5432'24", 

c = 11238'42". C = 10214'12 '. 

(,/) a - 70() / 37", W A - 17217 / oG", 

ft - 12530'52", B = 828'20", 

c - 6347'55". C = 423'35". 

4. Solve the polar triangles of the triangles of Exercise 2. 



300 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

5. Derive the following equations from (22) to (34) : 

cos j A cos ?B sin s 

sin ?C sin c 

cos vA sin ?B sin (s a) 

cos iC sin c 

sin jA cos ?B sin (s b) 

cos ^(7 sin c 

sin ^A sin ^# sin (s c) 

sin ?C sin c 

6. Prove that the following relation holds true for a right spherical 
triangle : 

tan 2 ^A = sin (c 6) esc (c + 6). 

148. Napier's analogies. This article is devoted to deriving 
formulas that may be used to solve triangles for which the given 
parts are two sides and the included angle or two angles and the 
included side. Substituting A = ^A and B = ^B in (7) and 
(10) of 53, we get 

sin %(A + B) = sin ^A cos %B + cos 4 sin \B, (36) 
sin %(A B) = sin ^A cos %B cos ^A sin %B. (37) 

Dividing (37) by (36) member by member, we get 

sin ^(A B) sin ^A cos %B cos ^^1 sin ^B 

sin %(A + B) ** sin %A cos %B + cos %A sin %B ( ^ 



Or, dividing both numerator and denominator of the right-hand 
member of (38) by sin \A sin ^5, 

sin $(A - B) cot^4 - cot %B 



B) 

and (33) we 
^- - 



From (31) and (33) we find cot \A = sm "" a ^ anc j cot ^ B = 



we obtain 



Substituting these values in (39) and canceling r, 



sin ^(A B) __ sin (s a) sin (s b) 

sin \(A + B) "" "sin (s - a) + sin (s - 6)' ( ' 



148] NAPIER'S ANALOGIES 301 

Using (34) and (33) of 57 to transform the right-hand member 
of (40), we get 



sin %(A - B) _ _2 cos %(2s - a - b) sin %(b - a) 

sin $(A + B) 2 sin %(2s - a - 6) cos (6 ^7)' ( ' 



Replacing (2s a 6) by c in (41) and simplifying slightly, we 
get 

ten iCfl &) 

' l } 



sin (A + B) tan c 

Again, using (11) and (8) of 53 with A = %A and B = 
we get 



cos ^(A B) = cos ^-A cos \E + sin ^-A sin ^B, (43) 
cos ^(A + B) = cos ^-A cos %B sin ^A sin %B. (44) 

Dividing (43) by (44) member by member, then dividing numer- 
ator and denominator of the right-hand member of the resulting 
equation by sin ^-A sin ^B and finally replacing cot ^A by 

sin (s - a) . , , , sin (s - 6) 

- - - - and cot %B by - - - > we have 

sin (s a) sin (s 6) 

~ 



cos -^-(A + B) sin (s a) sin (s b) 



Replacing r 2 by its value from (32) and simplifying slightly, we 
obtain 

cos i(A B) sin s + sin (s c) /J/SX 

- ^ - = - (46) 

cos ^(A + B) sin s sin (s c) 

Treating the right-hand member of this equation in a manner 
similar to that employed in transforming (40), we get 



cos (A +B) tan c 

Applying (42) and (47) to the polar triangle, we obtain 



302 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 



sin-(a + b) cot 

cos ^(a - &) tan 



COS J(ff + &) COt iC 



^ ^ 



The formulas (42), (47), (48), and (40) are known as Napier's 
analogies. These formulas are analogous to the law of tangents 
in plane trigonometry. 

EXERCISES 

1. Apply (42) and (47) to the polar triangle, then proceed in a 
manner analogous to that pursued in this article and obtain formula* 
(48) and (49). 

2. Use formulas (42), (47), (48), and (40) to prove the following 
formulas known as Gauss's equations or Delambre's analogies. 



COS .jC 

, sin v(a b) 

sin -$(A B) = - . I " cos ^C. 
sin .jc 

i, 4 . m cosl(a + b) . l 
cos ^(A + 7^) = i sin iC, 
cos -$c 



cos - = - -! sin . 
sin s^r 

3. Show that the second of Gauss's equations can bo written 



- 

hav c 

4. From formula (47), show that in any spherical triangle one-half 
the sum of two angles is in the same quadrant as one-half the sum of the 
opposite sides; that is, \(a + b) and \(A + B) arc in the same quadrant. 

5. (a) Divide sin \(A B) sin -$A cos -g# cos i.l sin ^li by 
cos -%(A 1$) cos ?A cos ?B + sin J.1 sin %B, member by member, 
then proceed in a manner similar to that employed in this article in 
deriving (42) and thus deduce formula (48). 

(6) Derive formula (19) by dividing sin J(.l + .#) by cos J-(A + B). 

6. (a) Divide sin ^(A B) by cos^(/l + B) and proceed in a 
manner similar to that outlined in 5 (a) and derive the formula 

sin y(A - B) sin i( - b) . . 

-- if A i P\ = ---- 1/" "T\\ cot ^ c cot ^^* 
cos ^(A + B) cos i(a + 6) 



149] 



CASES III AND IV 



303 



149. Cases III and IV. Given two sides and the included 
angle or given two angles and the included side. The four 
formulas (42), (47), (48), and (49) are used to solve a triangle 
when the given parts are two sides and the included angle, or 
two angles and the side common to them. If the law of sines 
is used to find the last unknown after two unknowns have been 
found, often the ambiguity arising may be removed by using 
the theorem that states that the order of magnitude of the sides of 
a spherical triangle is the same as that of their respective opposite 
angles. 

Other sets of formulas may be obtained from (42) and (47) to 
(49) by the interchange of letters. For example, another set 
would result from replacing a by c, c by a, A by C, and C by A in 
(42) and (47) to (49). 

Example. Find A, B, and c for a spherical triangle in which 
a = 5756 / 53 // , b = 13720'33", C = 9448'6". 

Solution. In this example (6 - a) = 3941'50", (6 + a) = 
9738'43", %C = 4724 / 3 // . Formulas (48), (49), (42), and (47) 
may be written in the respective forms 



tan %(B - A) = sin %(b - a) esc (b + a) cot C, (48 ; ) 

tan (A + B) = cos %(b - a) sec %(b + a) cot C, (490 

tan %c = tan (6 - a) sin %(B + A) esc %(B - A), (42') 

tan \c = tan (6 + a) HOP \(B - A) cos \(B + A). (47') 

The following form indicates the computation. The number in 
parenthesis above each column refers to the formula associated 
with the column. 



(48') 



(49') 



(42') 



check (47') 



1(6 - a) = 3941'50" 
j(6 +a) = 97 C 38'43" 
\C - 4724'3" 
IB - A) = 3039'2" 
J(# + A) = 10038'58" 


2 sin 9 80531 
I CHC 00388 
I cot 9 96356 


I cos 9 88617 
Zsec (-)O 87602 
I cot 9 96356 


I tan 9 91915 

Zcsc 0.29260 
I sin 9 99245 


Uan (-)O 87214 

1 sec 06535 
I cos (~)9 26670 


I tan 9. 77275 


I tan (-)O 72575 



}c = 5759'56" 

A - 6959'56" B 



I tan 20420 I tan 20419 



c = 11589'52". 

These results could have been checked by the law of sines. 



304 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

EXERCISES 
1. Solve the following spherical triangles: 



(a) a = 30, 


(c) a = 30, 


(e) B = 30, 


B = 45, 


C = 150, 


a = 45, 


c = 60. 


b = 135. 


C = 60. 


(6) b = 135, 


(d) A = 150, 


(/) A = 60, 


A = 45, 


c = 30, 


6 - 120, 


c = 60. 


B = 120. 


C = 150. 



2. In the following triangles where two values for a part are given, 
select the proper value. 

(a) A = 65 13', B = 4928', 13033', C == 128 16', a = 8824 / , 

6 = 5648', c = 120ir. 

(b) A = 50 10', /* = 1355', C = 5()30', a = G935', 11025', 

6 = J2030', c = 7020'. 

(c) A = 12740', # = 4515', </ = 12442', 1520 ; , a = 6853', 

6 = 56 50', c = 1810'. 

(d) A = 5220', B = 4515', C = 12442', a = ()853 r , 6 = 5660', 

c = 10419', 1810'. 

3. Using Napier's analogies, solve the following spherical triangles: 

(a) c=H60'0", (d) a = 8618 ; 40", 
A = 700'0", 6 = 453tt'20", 

B = 13118'0". C = 12046'30". 

(6) a = 8837 / 40' / , (c) a = 416'0", 
c = 12518'20", b = 11924 / // , 

5 = 10216'36 /r . C = 1()222 / 30 // . 

(c) a = 7624'0", (/) c = 12018'33 // , 
6 = 5819'0 // , A = 2722'34", 

C = 11630'0". # = 9126'44". 

4. In the following spherical triangles, find the angles by means of 
Napier's analogies and the required side by using the law of sines. 

(a) a = 4245'0", (6) a = 13115'0", 

6 = 4715 / 0", 6 = 12920'0", 

C = 1111'41". C = 10337 / 23 // . 

150. Cases V and VI. Two of the given parts arc opposite^. 
Double solutions. For convenience of reference, a theorem from 
solid geometry is repeated here. 



150] CASES V AND VI 305 

Theorem. The order of magnitude of the sides of a spherical 
triangle is the same as that of their respective opposite angles. 
Or if a and b are a pair of sides of a spherical triangle and A and 
B the respective opposite angles, we know that if 

a < 6, then A < B. (50) 

When the given parts of a spherical triangle are two sides and 
an angle opposite one of thorn, say, a, 6, and .4, the angle B may 
be found by using the law of sines, 

D ^ & ' \ /ci\ 

sin B = - . sin A. (51) 

sin a ' 

Since sin B does not exceed 1 in magnitude, log sin B does not 
exceed zero. Hence no solution will exist when log sin B > 0. 

When log sin B < 0, a positive acute angle and its supplement 
must be considered for B. Each value of B must be consistent 
with (50). Hence, there will be no solution, one solution, or two 
solutions according as (50) is satisfied by neither, by one and 
only one, or by both of the values of B obtained from (51). If 
b = a, then B = A, and there is one solution. 

Accordingly, begin the solution of a spherical triangle in which 
a, b, and A are the given parts by using (51) to find log sin B. 
If log sin B > 0, there is no solution. If log sin B < 0, find 
two values of B, one a positive acute angle and the other its 
supplement. Then, to find c and C, use the given parts with 
each value of B that satisfies (50) in 

4 l sin (A + B) ^ 



sin } 2 (a 6) 

These formulas were obtained by solving Napier *s analogies (42) 
and (48) for tan ^c and cot ^C, respectively. 

A similar discussion, with the roles of sides and angles inter- 
changed, applies when the given parts are two angles and a side 
opposite one of them; (51) solved for sin b would first be used and 
then (52). 

Example. Given a = 5245'20", 6 = 7112'40", A = 4622'10", 
find c, B, C. 



306 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

Solution. Two solutions are to bo expected. First using 

sin B = sin 6 sin A esc a (!') 

to find B f and afterwards using (42') and (49) to find c\> C\ y Cz y 
and Cz y we obtain the solution indicated below. 





0') 


a 


= 5245 ; 20" 


1 c-sc 0.09906 


b 


= 7112'40" 


I sin 9.97622 


A 


= 4622'10" 


I sin 9.85962 


/B, 


= 5924'22" I sin 9.93490 


\B Z 


= 12036'38" 




(42') (49) 


?(B 


i - A) = 631'6" 


I esc 0.94492 


?(B 


L + A) = 5253 / 16 ;/ 


Zsin 9.90171 


/tan 0.12112 


t( 


b - a) = 913'40" 


I tan 9.21 075 


/ sec 0.00565 


t( 


b + a) - 6159'0" 




I cos 9.67185 


ici 


= 4846'26 // I tan 0.05738 




Ci 


= 9732 / 62" 




i-Ci 


= 5749'56" I cot 9.79862 


Ci 


= 11539 / 62' / 




(42') (49) 


*< 


6 - a) = 913'40" 


Han 9.21 075 


/ sec 0.00565 


*< 


6 + a) = 6l59'0" 




/ cos 9.67185 




2 - ^l) = 376'44" 


/esc 0.21941 




%(B 


2 + A) = 8328'54" 


Jsin 9.99718 


/tan 0.94211 


C 2 


= U58'35" I tan 9.42734 




C 2 


= 2967'10 // 




C 2 


= 1330'4" I cot 0.61961 


C 2 


= 270'8" 



This solution may be checked by the law of sines. 

EXERCISES 

Solve the following spherical triangles: 
1. a = 6852'48", 2. a = 340'30", 

6 = 5649'46", A = 6129'30", 

B = 4515'12 /; . B = 2430'30". 



151] MISCELLANEOUS EXERCISES 307 

3. a = 4215'20", 4. a = 5928'27", 
A = 3620'20", A = 5250'20", 

# = 4630'40". B = 667'20". 

6. b = 80, 6. a = 6329'56", 
A = 70, 6 = 13214'23", 

B = 120. C = 6118'27". 

151. MISCELLANEOUS EXERCISES 

Solve the following spherical triangles: 

1. a = 12022 / 40 // , 6. a = 405'26", 
6 = 11134'27", 6 = 118 <; 22'7", 
c = 9628'35". C = 1601'23". 

2. a =- 41VO", 7. b = 15017 / 26 // , 
6 = 11924 / 0' / , A = (U37 / 53 // , 
T - 4854 / 38 // . /^ - 13954'34". 

3. A - 12132'41", 8. a = 31ir7", 
/y = 82f>2'53", b = 32 19' 18", 
T - 9851 r 55". c - 3315'2l". 

4. f = 8(i15'15", 9. A = 0357 / 39 // , 
A = 15317'G", B = 3r)4 ; 3", 

B = 7843'32". r = 13244 / 8 // . 

6. 6 = 8421'56' f > 10. A = ^SS'IO", 

A = 115 3()'4,5", B = SS^tt'SO", 

B = 80 19' 12". T = 5955'10". 

11. In a spherical triangle given r, A , a + 6, derive 

, i 4 , i sin ( - c) 
tan ?A tan ^7^ = ift g - 

12. Given two sides and the sum of the opposite angles of a spherical 
triangle derive a formula from Gauss's equations (Kxercise 2, 148) 
for computing the remaining angle. 

13. Prove the relation 

cot a sin b cot A sin C + cos C cos b. 

Hint. Multiply equation (13) by cos 5, substitute in (11), and then 
divide by sin b sin a, etc. 

14. If ci and c 2 be the two values of the third side when A, a, b are 
given and the triangle comes under Case V, show that 

tan ^Cj tan ^c 2 = tan 1(6 a) tan ^(6 + a). 



308 THE OBLIQUE SPHERICAL TRIANGLE [CHAP. XIV 

15. If 6 is the base of an isosceles spherical triangle and if the equal 
sides a, c be bisected by the arc h of a great circle, show that 

sin -$h $ sin -%b sec \a. 

16. Prove that 

sin (s a) -\- sin (s 6) + sin (s c) sin s = 4 sin \a sin -%b sin Jc. 

17. In a spherical triangle A = B = 2(7, show that 

8 sin 2 gC'(cos s + sin ^-C) cos ^c = cos a. 

18. Show that 

sin %K sin (A - %E) 

hav a = . .. . 

sin B sin (7 

where K = (2er - 180) ando- = ? 2 (A + ^ + O- 

19. In an equilateral spherical triangle, show that 2 cos ^a sin ^A =* 1. 

20. If in a spherical triangle C = A + B, show that 

cos C = tan ^a tan -J-6. 

21. If the sum of the angles of a spherical triangle is 300, show that 

cos 2 ^a + cos 2 ^6 + cos 2 \c = 1. 



CHAPTER XV 

VARIOUS ME* HODS OF SOLVING OBLIQUE SPHERICAL 

TRIANGLES 

152. Introduction. In this chapter we shall again consider 
methods of solving triangles corning under the six-case classifica- 
tion of 145. The principal method of this chapter will consist 
in dividing the given triangle into two right triangles and apply- 
ing Napier's rules to the parts. 

153. Cases III and IV. Consider the solution of the spherical 
triangle in which the given parts are a, &, and C, that is, two sides 
and the included angle. Figure 1 represents the spherical 

A 

A 




triangle ABC with arc AD drawn perpendicular to side EC and 
with the given parts a, 6, and C encircled. Figure 2 represents 
the two right triangles of Fig. 1 drawn separately and prepared 
for the application of Napier's rules. By the regular procedure, 
we obtain from triangle I 

tan (p tan b cos C, (1) 

cot 6 = cos b tan C, (2) 

sin p = sin b sin C, (3) 

sin p = cot tan (p. (Check) (4) 

After <p, and p have been found by means of (1), (2), and (3), 
the parts p and *p' = a <p in triangle II will be known. Now 
apply Napier's rules to obtain the following formulas for solving 
triangle II: 

309 



310 VARIOUS METHODS OF SOLVING [CHAP. XV 

?' = a - <p, (5) 

cot B = cot p sin <p', (6) 

cot 0' = sin p cot ^', (7) 

cos c = cos p cos <p', (8) 

cos c = cot 0' cot , (Check) (9) 

A = + 0'. (10) 

If the given parts are not named a, b, and C, the computer 
may derive a new set of formulas, or he may obtain the desired 
set by interchanging letters in (1) to (10). For example, if 
the given parts are a, c, and B, get the appropriate formulas by 
replacing b by c, and c by b, B by C, C by B in (1) to (10). Thus, 
from (1), (2), and (3), we get 

tan <p = tan c cos B, 
cot = cos c tan B, 
sin p = sin c sin B. 

To solve a triangle when the two angles and the side common 
to them are known, use (11) of 139 to find two sides and the 
included angle of the polar triangle, solve the polar triangle 
by formulas (1) to (10), and from the result get the desired 
solution by again using (11) of 139. Also, one may drop a 
perpendicular from the vertex of one of the given angles to the 
opposite side and solve the two resulting right triangles by the 
methods of Chap. XIII. 

164. Observations and illustrative example. One can usually 
draw a rough sketch representing the spherical triangle under 
consideration and showing its associated pair of right triangles in 
their proper relative positions. He can then solve the two right 
triangles and assemble the desired solution from the computed 
parts. 

However, by keeping in mind the following observations, he 
may use formulas (1) to (10) without reference to a figure. 

(A) Each of the parts a, 6, c, A, B, C of a spherical triangle is 
positive and less than 180. 

(B) When tan <p is positive, <p should be chosen positive and acute. 
When tan <p is negative, <p should be chosen in the second quadrant. * 

* <p might be taken negative. The remaining part of the solution would 
have to be carried out in harmony with this choice. 



154] OBSERVATIONS AND ILLUSTRATIVE EXAMPLE 311 

(C) In accordance with Rule A, 136, p and C are of the same 
quadrant if *p is positive. 

(D) Each of the pairs <p and 6, <?' and 0', must be of the same 
quadrant and have the same sign. Thus, if <p f is negative and acute, 
0' must be negative and acute; if <p is positive and of the second 
quadrant, must be positive and of the second quadrant. 

(E) Angle B obtained from (6) is of the first or second quadrant 
according as cot B is positive or negative. It is not necessarily of 
the same quaiirant as p. 

The following solution will illustrate the application of these 
observations and the general method of procedure. 

Example. Solve the spherical triangle in which a = 7843 / , 
b = 11812', C = 5927'. 

Solution. The following form, showing the solution by means 
of formulas (1) to (10) of 153, is self-explanatory. 



(1) and (check) 



(2) 



(3) 



a - 7843' 
b = 11812' 
C = 5927' 


/tan (-)O 27068 
Zcos 9 70611 


Zcos (-)9 67445 
1 tan 22899 


/ sin 9.94513 
J sin 9 93510 


v = 13631'48" 


J tan (-)9 97679 






= 12840'55" 


I cot (-)9 90344 


/cot (-)9 90344 




p = 4922'27" 

j = -(5748'48") 
p - 4922'27" 


I sin 9 88023 

(6) and (check) 
I sin (-)9 92753 
/ rot 9 93343 


(7) 
/cot (-)9 79894 
/ sin 9 88023 


/ sin 9 88023 
(8) 
/ cos 9 72647 
/cos 9 81366 


B 12668'61" 
0' = -(6427'56") 


I cot (-)9 86096 
I cot (-)9 67917 


7cot"(-)9 6"79l"7 




c = 6942'21" 


/cos 9.54013 




/ cos 9 54013 



e + 0' ** 6412'59". 



Figure 3 shows the right tri- 
angles CAD and DBA in their 
proper relative positions. 




312 



VARIOUS METHODS OF SOLVING [CHAP. XV 



EXERCISES 

Solve each of the following triangles by solving the two auxiliary 
right triangles: 



1. C = 1295'28", 
B = 14212'42", 
a = 604'54". 

See Fig. 4. 




Solve 

3. a 
b 
C 

4. 6 
c 

A 

5. a 
b 
C 



FIG. 5. 

the following spherical 

= 8824 / // , 

= 5648'0", 

= 12816'0". 

= 12030'0", 

= 7020'0", 

= 5010'0". 

= 7624 / 0' / , 

= 5819'0 ;/ , 

= IIG^O'O". 



2. A = 3134'26", 

B = ao^s'i^', 

c = 702'3". 
See Fig. 5. 



triangles by the method of this article: 

6. a = 8837'40", 
c = 12518'20", 

B = 10216 / 36". 

7. a = 8618 ; 40", 
6 = 4536'20", 
C - 12046'30". 

8. 6 = 13217'30 // , 
c = 7815 / 15' / , 

A = 4020 / 10". 



Solve the following triangles by solving the polar triangle. 



9. A 
B 
c 



= lOO'WO" 
= 305'0". 



10. A 

c 
b 



2722'34", 
9126'44 ;/ , 
12018'33 /; . 



155. Case III. Alternate method. Another set of formulas 
sufficient to solve the spherical triangle for which two sides and 



155] 



CASE III 



313 



the included angle are known do not contain p. Applying 
Napier's rule to triangle I of Fig. 6, we obtain 



Also 



tan (p = tan b cos C. (11) 



(12) 



A A 



Again, by using Napier's rules, we 
obtain from triangles II and I 



sin <p 
sin <p 



- cot B tan p, 
cot C tan p. 



(a) 



Dividing the first of these equa- 
tions by the second, member by 
member, and solving the result for cot #, we get 

cot B = cot C sin <?' esc <p. 




Fio. 6. 



(13) 



Note that the equations (a) were found by using #,', p, and B in 
triangle II and the homologous parts <p, p, and C in triangle I. 
The procedure to get (13) will be followed to obtain a formula 
for cos c. From triangles II and I, we get 



cos c = cos <p' cos 



cos b = cos <p cos p. 



Dividing the first of these equations by the second, member by 
member, and solving for cos c, we get 



From triangle I 
from triangle II 
and 



cos c = cos b sec <p cos 
cot B = cos b tan C; 
cot 0' = cos c tan 5, 



(14) 
(15) 
(16) 
(17) 



A = 6 + 8'. 

The law of sines may be used as a check formula. 

The observations of 154, except those referring to p, apply 
also to the solution based on the formulas of this article. 

Example. Use formulas (11) to (17) of this article to solve the 
spherical triangle in which a = 6820'25", b = 5218'15", 
C - 11712'20". 



VARIOUS METHODS OF SOLVING 



IC'HAP. XV 



Solution. The solution and the chock by the law of sines are 
displayed in the following form : 

(11) (13) (14) (15) (16) 



a = 6820 / 25" 

6 = 5218'15" I / tan 11194 

C=11712'20"[|cps (-)9 66009 

* = 14923'29" Jtan(- 

j = _ p = -813'4" 

B = 454'41" 



77203 



Jcot(-)9 71100 
1 csc 29314 
[ls\n (-)9 99468 
/ cot 9 99882 



I cos 9 78638 



I sec (-)() 06517 
/cos 9 19188 



-)9 04343 



/ cos 



9 78638 



I tan (-)O 28900 



e = 139W51" 

tf = -834(X35 // 
.4 - 



Jcot (-)O 07538 



I tan 00118 
/cos(-)9 04343 



Jcot(-)9 04461 



Check a I / sin 9 96820 

.1 1 1 sin 9 91995 

04825 



b \l sin 9 89832 
B I sin 9 85008 



c j J sin 9 99733 
('[/sin 9 94909 
04824 



04824 

EXERCISES 

Solve the following spherical triangles by the method of this article. 
l.o = 8824'()'', 4. a = 8837 / 40", 



b = 5648'0 ;/ , 
C = 128 P 16 ; 0". 

2. b = 12030'0", 
c = TO^O'O", 

A = SO^O'O". 

3. a = 7624'0", 
6 = 5819'0", 
C = 11630'0 ; '. 



c = 12518'20", 
B = 10216'36". 

5. a = 8618'40 r ', 
6 = 45 36'20", 
C = 12046'30". 

6. b = 13217 / 30' / , 
c = 78 15 ; 15", 

^1 = 4020'10". 



Solve the following triangles by solving the polar triangle. 

7. A = 12010'0", 8. A = 2722'34 / ', 

5 = lOO^O'O", C = 91 26 ; 44", 

c = 305'0". b = 12018'33 // . 

9. Using Fig. 7, derive formulas (a) to (g). 



cot 6 = cos c tan B, 

B' = A - 6, 

tan b = tan c cos sec 0', 
cos (7 = cos B csc sin 0', 
tan ^ = cos B tan c, 
tan ^' = cos C tan 6, 
a == ^ -(- ^' t 




(a) 
(6) 
(c) 
(d) 
(e) 
(/) 
(0) 



Using the formulas of Exercise 9, solve each of the following triangles : 



156] HAVERSINE SOLUTION OF CASE III 315 

10. a = 1295'28", 11. A = 3134'26", 

B = 14212'42", B = 3028'12", 

C = 604'54". c = 702'3". 

166. Haversine solution of Case III. Evidently the law of 
cosines could be used to find a when 6, c, and A are given. This 
would not, however, be convenient for logarithmic computation. 
A formula for finding a directly by usin^ a table of haversines 
will be developed from the law of cosiro-i. 

The law of cosines may be written 

cos a = cos b cos c + sin b sin c cos A. (18) 

By definition hav 8 = ^(1 cos 6). Solving this for cos 0, we 
get cos e = 1 2 hav 0. Hence 

cos a = 1 2 hav a, cos /I = 1 2 hav 4. (19) 

Substituting the expressions for cos a and cos A from (19) in (18), 
we obtain after slight simplification 

12 hav a = cos b cos c + sin b sin c 2 sin b sin c hav A. 

(20) 

Now cos b cos c + sin 6 sin c = cos (b r) = 1 2 hav (b c). 
Replacing cos b cos c + sin b sin r by 1 2 hav (/> r) in (20) 
and solving for hav a, we obtain 

hav a = hav (b c) + sin b sin c hav A. (21) 

Similarly, 

hav b = hav (a c) + sin a sin c hav $, (22) 

hav c = hav (a h) + sin a sin b hav C. (23) 

After a side has been computed by the haversine formula, three 
sides and an angle will be known. The other two angles may 
then be obtained by using the law of sines. The facts that when 
a < b < c then A < B < C and that the sum of two sides is 
greater than the third side will often serve to determine the 
quadrant of each angle thus found. Also a rough sketch will 
sometimes serve the same purpose. When the quadrants of the 
angles cannot be determined by the methods suggested, other 
formulas should be used. For this purpose, the result of solving 
(21) for hav A, 



316 VARIOUS METHODS OF SOLVING [CHAP. XV 

(24) 
and the corresponding formulas for hav B and hav C are useful. 



, A hav a hav (6 c) 

hav A = : = r-^ -, 

sin 6 sin c 



Example. Use (21) to find the side a of a spherical triangle in 
which b = 5929'30", c = 10939'40", A = 5010'10"; then find 
B and C by the law of sines. 

Solution. The formulas to be used are 



hav a = hav (b c) + sin b sin c hav 
sin B = sin 6 sin A esc a, 
sin C = sin c sin A esc a. 

The solution is displayed in the following form: 



(a) 
(6) 
(c) 





(a) 


(a) (6) (c) 


b - 


5929'30" 


I sin 9 93529 




I sin 9. 93529 


c = 


10939'40" 


lain 9.97391 






I sin 9. 97391 


A = 


5010'10" 


2 hav 9.25465 




I sin 9 88533 


lain 9.88533 


c - 6 - 


5010'10" 




1 n hav 17974 
1 n 14583 






log 9.10385 


a = 


6934'56" 


n hav 32557 


1 csc 02818 


1 esc 02818 


B - 


4454 / 36" 


I sin 9 84880 




C = 


12929'54" 


I sin 9. 88742 



b = 13214'23", 

c = eris^?". 



EXERCISES 

Using the haversine formula, find the unknown side in the following 
spherical triangles: 

1. b = 1258', 3. a = 6329'56", 
c = 6426', 

A = ioo4'. 

2. a = 131 15', 4. C = 4820', 
b = 12920', 6 = 5210 / , 
C = 10337'20". a = 4920'. 

6. Solve Exercise 3 for B and A by using the law of sines. 
6. Using the relation cos B = 1 2 hav 0, derive from the cosine law 
hav c = hav (a - b) hav (180 - C) + hav (a + b) hav C. 

157. Cases V and VI. Consider the solution of the spherical 
triangle in which the given parts are a, 6, and A. In this case 
there may be two solutions. Figure 8 represents the spherical 
triangle ABC with arc CD drawn perpendicular to side AB and 
with the given parts A, a, and b encircled. The dotted line 
indicates a second position that arc CB may assume. 



157] 



CASES V AND VI 



317 





FIG. 9. 



To obtain the formulas for solving a spherical triangle in which 
a, 6, and A are the given parts, apply Napier's rules to triangle I 
in Fig. 9 to obtain 



tan <p = tan b cos A, 

cot 9 = cos b tan A, 

sin p = sin b sin A, 

sin p = tan <p cot B. (Check) 



(25) 
(26) 
(27) 
(28) 



Since p is found from (27), p and a will be known in triangle II 
after triangle I has been solved. Hence apply Napier's rules to 
triangle II to get 



' = 



cos <?' = cos a sec p, 

sin B = esc a sin p, 

cos B' = cot a tan p, 

cos 0' = cos <?' sin J5. (Check) 



Also it appears from Fig. 8 that 
c = <? + ?', 



(29) 
(30) 

(3D 
(32) 

(33) 
(34) 



The interchange of certain letter pairs in formulas (25) to (34) 
will give a new set of formulas applicable to a triangle for which 
the given parts are denoted by other letters than a, 6, and A. A 
spherical triangle for which two angles and a side opposite one 
of them are given can be solved by applying formulas (25) to 
(34) to its polar triangle. Also a perpendicular may be drawn 
from the vertex of the unknown angle to the opposite side and 
special formulas derived by means of Napier's rules. 



318 



VARIOUS METHODS OF SOLVING 



[CHAP. XV 



158. Observations and illustrative example. Slight modifica- 
tions of the observations made in 154 apply to the solution under 
consideration. Since the cosine of a negative angle is the same 
as the cosine of an equal positive angle, two values of <p' ', one the 
negative of the other, are chosen, and the solution corresponding to 
each value is formed. 

Since B is found from its sine, an angle and its supplement are 
written. From triangle II, cot B = cot p sin <p'. Therefore B 
is of the same quadrant as p when <p' is positive. If y f is nega- 
tive, B is of the first or second quadrant according as p is of the 
second or first quadrant. 

If cos (p' = 1, <p f = 0, and there is only one solution. If 
log cos (p 1 > 0, there is no solution. Also each of the quantities 
b and B found from (33) and (34) must not be negative nor 
greater than 180. Hence no solution corresponds to a value of 
<p' if either of the quantities v + <p' or + tf is greater than 
180. 

The following solution will illustrate tho method of procedure. 

Example. Solve the spherical triangle* in which A = 11512', 
b = 7310', a = 
Solution. 



b = 7310' 
A = 11512' 
V = 12523'51" 
= 12136'30" 

p 11959'43"* 

p - 11959'43" 
a = 11035' 
>' = (4518'46") 
B - 11218'48", 6741 


(25) and (check) 
1 f tan 51920 
\l cos (-)9 62918 


(26) 
I cos 9 46178 
I tan (-)IO 32738 


(27) 
/ sin 9 98098 
I sin 9 95657 

/ sin 9 93755 
(31) 
I tan (-)O 23864 
/cot (-)9 57466 


I tan (-)O 14838 
I cot (-)'.) 78916 


I cot (-) 9.78916 

(30) 
1 / sin 9 93755 
\l CBC 02865 

I sin 9. 96620 


I sin 9 93754 
(29) and (check) 
I/ sec (-)O 30109 
|Zcos(-)9 54601 


I cos 9 84710 
'12" I sin 9 96620 



0' - (4924'52") / cos 9 81330 

c - * *' - 17042'37" and SO^VS" 

C = e tf - 171l / 22" and 7211'38" 

Therefore the two solutions are 

ci = ITiPia'ST", Ci = 171l / 22" 
02 - 805'5", C 2 = 7211'38" 



I cos 



9.81330 



11218'48",t 
6741'12". 



* p was chosen in the second quadrant in accordance with Rule A of 136. 

t B = 1 1218'48" was placed in the solution associated with the positive 
value of <p' and B' in accordance with the observation in the second para- 
graph of this article. 



159] CASES I AND II 319 

EXERCISES 

Solve the following spherical triangles by the method of this article. 

l.o = 406'0", 3. a = 15057'5", 
b = 11822'0", b = 13415'54", 

A = 29 43'0". A = 14422 ; 42". 

2. a = 12815'0", 4. a = 5245'20", 
6 = 12920'0", c = 7J12'40", 

A = 13025'0". A = 4l522'10". 

6. Solve each of the following triangles by solving its polar triangle. 

(a) c = 8013'0", (b) a = 11513'4", 

C = 7815'0", A = 12043'0", 

B = 7517'0". B = 11638'0". 

6. Solve each of the following triangles by dropping a perpendicular 
from the unknown angle to the opposite side and solving the right 
triangles formed. 

(a) a = 15042'40", (b) a = 14712'40", 

A = 145 52'10", A = 14212'10", 

C = 79 37 r 20". B = 7557 / 20 // . 

7. Using Fig. 10, derive formulas (a) to (g) of this exercise. 

C 

tan (p = cos A tan 6, 
cos <p' = cos <p cos a sec 6, 

c = <p + <?', 

tan B = tan A esc <?' sin < 
cot = cos 6 tan A , 
cot 6' = cos a tan #, 
C = e + 6'. 

FIG. 10. 

8. Using the formulas of Kxercise 7, solve Exercises 1 to 3. 

159. Cases I and II. The most expeditious method of solving 
a spherical triangle in which three sides arc given employs 
formulas (31) to (34) of 146. However, one angle may be 
found by using 

cos A = (cos a cos 6 cos c) esc 6 esc c, 




320 VARIOUS METHODS OF SOLVING [CHAP. XV 

a formula obtained from the law of cosines, or by using (24) of 
156, namely 

hav A = [hav a hav (b c)] esc 6 esc c. 

Two sides and the included angle will then be known, and the 
method of 153 may be employed. The spherical triangle for 
which three angles are given may be solved by means of its polar 
triangle. 

EXERCISES 

Solve the following spherical triangles: 

1. a = 57, 4. A = 11635 ; 36", 
b = 137, B = 10514'48", 
c = 116. C = 4317 ; 12". 

2. A = 150, 5. a = 7736'12", 
B = 131, b = 6316'48", 
C = 115. c = 10723'12". 

3. a = 14930', 6. A = 13619'36", 
6 = 1310', B = 4318'30", 
c = 11920'. C = 11443'18". 

160. MISCELLANEOUS EXERCISES 

Solve the following spherical triangles: 

1. a = 7624'40", 6. a = 9940'48", 
6 = 5818'36", 6 = 6423'15", 
C = 116 30 ; 28". A = 9538 / 4 // . 

2. 6 = 9940'48", 6. A = 7311 / 18", 
c = lOO^g'SO' 7 , B = 6118'12", 

A = GS^S'lO". a = 46 45'30". 

3. A = 3134 / 26", 7. a = 5717', 
J5 = Stm'l^', b = 2039', 

c = 702 / 3". c = 7622'. 

4. a = 405'26", 8. A = 8620 ; , 
b = 11822 / 7 // , 5 = 7630', 

A = 2942'34 // . C = 9440 / . 

9. A ship sailing on a great circle crosses the equator in longitude 
7826' W. with course 4332'. Find its latitude when its longitude is 
10 W. 

10. A ship sails 5400 nautical miles from San Francisco along a great 
circle with initial course of 24025 / . Find the position reached. (For 
San Francisco, longitude X = 12323 / W; latitude L = 3828' N.) 



160] MISCELLANEOUS EXERCISES 321 

11. Find the pole (L, X) of the great circle of Exercise 10. 

12. An airplane flies 7000 nautical miles along a great circle. If the 
initial course is 2532' and if it reaches a point in latitude 1815' N. 
and longitude 12 15' W., find the position of departure. 

13. Using (21) and (24), find the initial course and distance for a 
voyage along a great circle from Los Angeles (latitude L = 3403' N., 
longitude X = 11815' W.) to Auckland (latitude L = 4118' S., 
longitude X = 17451' E.). 

14. Using (24) find the three angles of the spherical triangle in 
which a = 7014'20", b = 4924'10", c = 3846'10". 



CHAPTER XVI 
APPLICATIONS 

161. Nature of applications. Many applications of spherical 
trigonometry deal with time and with angular distances. These 
considerations of time arid distance may have reference 1 to bodies 
far removed from the earth (celestial) or to bodies on the earth 
(terrestrial). 

The shape of the earth is approximately that of a sphere having 
a diameter of 7917 miles. In what follows we shall consider it as 
a sphere. Hence the problem of finding the great-circle distance 
between two points on the earth or of locating a point on it is a 
problem that, may be solved by the* use of spherical trigonom- 
etry. Time enters our considerations because the rotation 
of the earth about its axis once every day furnishes the basic 
unit of time. 

162. Definitions and notations. The earth revolves about a 
diameter called its axis. One point where the axis cuts the 
surface of the earth is called the north pole, P n ; the other is called 

the south pole, P 8 . 

The equator is the great circle on 
the earth whose plane is perpendi- 
cular to the axis of the earth. 

A meridian is a great circle on 
t he earth passing through the; north 
pole and the south pole. In Fig. 
1, P n BP a and P n CP a represent 
meridians. Since meridians cut the 
equator at right angles, angular 
distances of points on the earth 
from the equator are measured 
along meridians. 

The latitude (Lat. or L) of a point on the earth is the angular 
distance of the point from the equator. It is measured along a 

322 




163] 



COURSE AND DISTANCE 



323 




FIG. 2. 



meridian north or south of the equator from to 90. In Fig. 1, 
CM 2 represents the latitude of M 2 . In general, north latitude is 
considered positive, south latitude negative. 

Because of the great importance of triangle MiPJlf 2 in con- 
nection with problems relating to distances and angles on the 
earth, it is called the terrestrial tri- 
angle. Arc M\Mi represents the dis- 
tance along the great-circle track from 
Mi to Af 2 , and the angle M z MiP n 
gives the initial direction of the track. 
The angle of departure P n M\M% 
measured from the north around 
through the east from to 360 is 
called the initial course C n . For a 
person situated on the northern 
hemisphere of the earth at a point 
such as z in Fig. 2, north is along the 
tangent to the meridian away from the equator; for a person 
standing at z facing north, oast is on his right, west is on his left, 
and south is opposite to the direction in which he is facing. 

Figure 3 indicates directions at four positions on the earth. 

The longitude (Long, or X) of a point on the earth is the angle 
at either pole between the meridian 
passing through the point and some 
fixed meridian known as the prime 
meridian. It is measured east or 
west of the prime meridian from 
to 180. The meridian of Green- 
wich, England, is the prime meridian, 
not only for English and American 
navigators but also for those of 
many other nations. 

The latitude and longitude of a 
point give its position on the earth 
just as the two coordinates of a point give its position relative to 
a set of rectangular axes. 

163. Course and distance. In general, the procedure of apply- 
ing spherical trigonometry to solve problems relating to the earth 
consists in finding three parts of the terrestrial triangle, solving 



W 




324 



APPLICATIONS 



[CHAP. XVI 



for one or more of the other three parts, and interpreting the 
results. Consider, for example, the problem of finding the great- 
circle distance between two points MI and M 2 when the latitude 
and the longitude of each point are known. In Fig. 4, P n 
represents the north pole, QKiK^Q' the equator, P n GQP the 




meridian of Greenwich, and MI and M 2 two places on the earth. 
The longitudes Xi of MI and X 2 of M 2 are known; hence angle 

MiP n M z = X 2 - Xi 

is known. Also, the latitudes L\ = K\M\ of Mi and L 2 = K 2 M Z 
of M 2 are known; hence the arcs MiP n = 90 L\ = co-L\ 
and J/j&Pn = 90 Z/2 = c0-L 2 are known. Thus, in triangle 
MiP n Mz, two sides M \P n = co-Li and 
MzP n = co-L 2 and the included angle 
MiP n Mz = X 2 Xi are known. Con- 
sequently, we can solve this triangle by 
Napier's analogies, by the method of 153 
or by that of 156. 

Example. Compute the initial great- 
circle course and the distance for a trip 
f rom St. Augustine lighthouse I/ 1 = 30 N., 
Xi = 76 W. to the Strait of Gibraltar 
I0 ' ' L 2 = 36N.,X 2 = 5 30 ; W. 




163] 



COURSE AND* DISTANCE 



325 



Solution. Substituting from Fig. 5, 90 - Li for a, 90 - L 2 
for b, Xi X 2 for C, MI for B, and D for c in formulas (11), (12), 
(13), and (14) of 155, we obtain 

tan <p = cos (Xi X 2 ) tan (co-L 2 ) = cos (Xi X 2 ) cot L 2 , (a) 
9 ' = 90 - Li - <p = 90 - (Li + ?), (6) 

cot MI = cot (Xi X 2 ) sin <p f esc <p 

or cot MI = cot (Xi X 2 ) cos (L\ + (p) esc ^, (c) 

cos D = cos ^>' sec <p cos (co-L z ) = sin (Li + ^) sec ^ sin L 2 . 

(d) 

Substituting the given values in formulas (a), (6), (c), and (d) 
and evaluating <p, MI, and D from the results, we obtain the 
following solution: 



Xl - \2 

L 2 
1 


= 7030' 
= 36 
= 2440'35" 
= 5440'35" 
= N.6352'30' 
= 588'43" = 


(a) 
1 I cos 9 52350 
1 / cot 13874 


(c) 
I cot 9 54915 

I esc 37935 
I cos 9 76208 


(d) 

I sin 9 76922 
1 sec 04159 
I Hin 9 91163 


(Check)! 

Uan 0.14956 
I cos 9 64378 
1 tan 20666 
log 00000 


I tan 9 66224 

'E. 
3488.7 miles* 


/ cot 9 69058 


I C089.72244 



The problem of finding course and distance is conveniently 
solved by using formula (23) 156 to find distance D and then 
using the law of sines to find the course angle. To apply (23), 
156, to Fig. 5, replace c by D, a by 90 - LI, b by 90 - L 2 , and 
C by Xi X 2 to obtain 

hav D = hav (L 2 Li) + cos L\ cos L 2 hav (Xi X 2 ). (1) 



The law of sines applied to Fig. 5 gives 

sin MI = cos L 2 sin (Xi X 2 ) esc D. 



(2) 



So far as formula (2) is concerned the angle MI may be of the 
first quadrant or of the second. A navigator usually knows 
the course approximately and thus knows the quadrant to be 
expected. Very often the quadrant of MI can be determined by 
considering that the order of magnitude of the sides of a spherical 

* 1' of angle at the center of the earth subtends 1 nautical mile = 6080 ft. 
on a great circle of the earth. Hence, when an arc of a great circle on the 
earth is expressed in minutes, it is also expressed in nautical miles. 

t The check formula was obtained by drawing a perpendicular from M\ to 
P n M 2 in Fig. 5 and applying Napier's rules. 



326 



APPLICATIONS 



[CHAP. XVI 



triangle is the same as that of the opposite angles or by a rough 
sketch. When the suggested methods fail, the law of sines should 
not be employed. In such cases, the following formula may be 
used: 

hav A = [hav a hav (6 c)] esc 6 esc c. 

EXERCISES 



1. Figure 6 represents the terrestrial tri- 
angle with the arc of a great circle drawn 
through Mi perpendicular to P n M\. Apply 
Napier's rules to the figure to obtain 

tan (f> cos (X 2 Xi) cot L 2 , 

<?' = 90- (Li+rt, 
cos D = sin L 2 sec <p sin (L\ + ^>), 
cot C = cot (\2 Xi) esc (p cos (Li + <p). 




. 6. 



2. In formulas (11) to (14) of 155 substitute 90 - L t for a, 90 - L 9 
for 6, X 2 Xi, for C, M\ for B, and /) for c to obtain the formulas of 
Exercise 1. 

3. Substitute for a, ft, c, and C of formula (23) of 156 appropriate 
values from Fig. 6 to obtain 

hav D = hav (Li L> 2 ) -\- cos L\ cos L 2 hav (X 2 Xi). 

Then write a formula from the law of sines for finding the course 
angle M\. 

4. Substitute for a, 6, c, A, B, and C appropriate values from Fig. 6 
in formulas (42), (47), (48), (49) of 148 to obtain formulas for solving 
the triangle of Fig. 6 completely. 

6. Find the initial compass course and distance in nautical miles for a 
great-circle voyage from San Diego (Li = 3243' N., Xi = 11710' W.) 
to Hong Kong (L 2 = 229' N., X 2 = 11410' E.). Use the formulas of 
Exercise 1. 

6. The great-circle distance from Cape Flattery, 4824 ; N., 12444' 
W., to Tutuila, 1418' 8., 17042' E., is 5084.75 miles. Find the course 
of the ship on arrival at Tutuila if it follows a great-circle track from 
Cape Flattery to Tutuila. 

7. Find the distance by great circle from New York, LI = 4040' N., 
Xi = 4 h 55 m 54" W., to Cape of Good Hope, L 2 = 3356' S., X 2 = 
l h 13 m 55" E. 



163] COURSE AND DISTANCE 327 

8. The distance from Cape Flattery, 4824' N., 12444' W., to 
Tutuila, 1418' S., 17042' E., is 5085 miles. Find the initial course for 
a trip from Cape Flattery to Tutuila, by great circle. 

9. Find the initial course and the distance for a great-circle voyage 
from Cape of Good Hope 3422' S., 1830' E. to Singapore 117'30" N., 
10351' E. Also find the latitude and longitude of the northern vertex* 
(the most northerly point) of this great-circle track. Use the formulas 
of Exercise 3. 

10. Find the initial course and the distance for a voyage along a 
great circle from Los Angeles L = 3403' N., X = 11815' W. to Auck- 
land L = 4118' S., X = 17451' E. 

11. The northern vertex of the great-circle track from San Francisco, 
Lat. 3828' N., Long. 12323' W., to Manila, Lat. 1435' N., Long. 
12057' E., has Lat. 4607' N., Long. 16333'36" W. Find the latitude 
reached when the longitude is 180. 

12. The northern vertex of a great-circle track is in L = 6050'26" N., 
X = 6029'37" E. Given the following positions: 

Rio de Janeiro: L = 2255' S., X = 4309' W., 
Strait of Gibraltar: L = 3553' N., X = 542' W., 
Cape St. Roque: L = 529' S., X = 3515' W., 
Cape Manuel: L = 1439' N., X = 1727' W. 

When following this track, what will be the 

(a) Longitude when in the latitude of Rio de Janeiro? 

(b) Latitude when in the longitude of Gibraltar? 

(c) Longitude when in the latitude of Cape St. Roque? 

(d) Latitude when in the longitude of Cape Manuel? 

(e) Course and distance when in the latitude of Rio de Janeiro? 
(/) Distance from vertex when in the longitude of Gibraltar? 

13. A ship sails from San Francisco L = 3828'24" N.,X = 12322'54" 
W., to Manila L = 1435'48" N., X = 12()57'18" E., following a 
great-circle track. Find the course angle at departure, the course 
angle at arrival, and the distance traveled. 

14. Substitute 90 - Li for a, 90 - L* for 6, Xi - X 2 for C, MI for 
B, M 2 for A, D for C, in (42), (47), (48), (49) to obtain: 

sin !(Af 2 MI) tan -^(Lz L\) 



+- Mi) tan -$D 

* A meridian passing through the vertex of a great-circle track is per- 
pendicular to the track. 



328 



cos 



APPLICATIONS 

- Mi) cot i(Li + L 2 ) 



[CHAP. XVI 



cos 2 

sin -g(L 2 Li) 
cos ^(L 2 + Li) 
cos ^(L 2 Li) 



sn - 



Li) 



tan^D 
tan ^(M 2 MI) 

cot -j(Xi X 2 ) 
tail \(M i + M 2 ) 

cot 



X 2 ) 



Using these formulas, solve Exercise 8. 



164. The celestial sphere. Consider a fixed star so far away 
from our solar system that the light rays coming to us from this 
star appear to follow parallel lines independent of our position; 
for example, light rays coming from this star to us at one position 
of the earth's orbit appear to have the same direction as light rays 
coming from the star to us 6 months later when we are on the 
other side of the orbit of the earth or approximately 186 million 
miles from the first position. Since, to us, light rays from this 
star seem to travel in parallel lines, we naturally associate a fixed 
direction with it. 

We shall speak of the celestial sphere as a sphere concentric 
with the earth and having a radius of unlimited length; by this 
we shall understand that any two parallel lines cut this sphere in 
the same point, and any two parallel planes cut it in the same 

great circle. With any point on 
this sphere is associated a fixed 
direction; the angular distance 
between two points on it may be 
considered, but not an actual dis- 
tance in miles. 

Figure 7 represents the celestial 
sphere with the earth at its 
center. 

The point P N on the celestial 
sphere where a line connecting 
the center of the earth to its north 
pole cuts the celestial sphere is called the north celestial pole; the 
point P s diametrically opposite is called the south celestial pole. 
The plane of the equator of the earth cuts the celestial sphere 
in the equinoctial or celestial equator. The celestial poles are the 
poles of the celestial equator. 




Q 



FK. 7. 



164] 



THE CELESTIAL SPHERE 



329 



The groat circles such as P N MP K in Fig. 7, passing through the 
celestial poles, are called hour circles or celestial meridians. 

The point Z (see Fig. 8) directly above an observer, that is, 
the point where a line connecting the center of the earth to an 



Upper 
Transit 



North 

Celestial 

Pole 




North / ^--" -~^/\l '' gf~T---.Y \ South 

Point N fcl Earthy W VN p o int 

of " ^T Kor.- 7 A^7 v I> of 

Horizon \^^SaS2n_^_ / \T-^& ^1 Horizon 



South 

Celestial 

Pole 



Na 

Nadir 

FIG. 9. 

observer on it would intersect the celestial sphere, is called the 
zenith. The point on the celestial sphere diametrically opposite 
the zenith is called the nadir Na (see Fig. 9). 

The horizon NWSE of an observer is the great circle on the 
celestial sphere having the zenith and nadir as poles. A plane 



330 APPLICATIONS [CHAP. XVI 

tangent to the earth at a point on it intersects the celestial sphere 
in the celestial horizon associated with the point. 

The point on the horizon directly below the north celestial 
pole is called the north point of the horizon. The south point, 
the cast point, and the west point of the horizon are then deter- 
mined in the usual way. 

The great circles, such as ZMK of the celestial sphere, which 
pass through the zenith, are called vertical circles. Evidently they 
are all perpendicular to the horizon. The prime vertical is the 
vertical circle EZW (see Fig. 8) passing through the zenith and 
the east and west points of the horizon. 

Figure 9 exhibits both the equinoctial system and the horizon 
system. 

166. The astronomical triangle. The spherical triangle (see 
Kig. 10) whose vertices are the north celestial pole, the zenith, and 
the projection of a heavenli/ body on the 
celestial sphere is called the astronomical 
triangle. The solution of many of the 
problems of astronomy and of navigation 
requires the solution of this triangle. 

The great-circle distance of a point on 
the celestial sphere from the celestial equator 
is called the declination d of the point. This 
corresponds to the latitude* of a point on the 
earth. Inspection of Fig. 9 shows that the 
arc PxM of the astronomical triangle is 90 
minus declination, or co-d. 

The hour angle t of a point on the celestial sphere is the angle 
between the hour circle passing through the zenith of the observer 
and the hour circle passing through the point.* As the earth 
turns on its axis, the heavenly bodies appear to move on the 
celestial sphere. Thus the angle through which the earth must 
turn to bring the celestial meridian of an observer into coincidence 
with the hour circle of a point on the celestial sphere appears 
as the hour angle of the point relative to the observer. The 
significance of the word hour angle appears when we consider 

* Hour angle is often expressed as so many degrees east or west, according 
as the body observed is in the eastern sky or in the western sky. It is oftf v 
measured toward the west from 0* to 24* (360). 




166] 



(77 YEN I, </, L; TO FIND h AND Z 



331 



that the earth turns on its axis and moves in its orbit in such a 
way that the sun crosses the meridian of a place once every 
24 hours. 

The altitude h of a point on the celestial sphere is its great-circle 
distance from the horizon. Inspection of Fig. 9 shows that the 
:dde MZ of the astronomical triangle is 90 minus altitude 
or co-h. 

The azimuth Z n of a point on the celestial sphere is the angle at 
the zenith between the vertical circle of the point and the celestial 
meridian of the observer. It is usually measured from the north 
point around through the east from to 360. It is easy to write 
the azimuth Z n when the angle Z of the astronomical triangle has 
been found. 

Evidently the length 1\Z of the; astronomical triangle is 90 
minus the latitude of the observer, or 90 L. 

166. Given /, d, L\ to find h and Z.* Figure It represents the 
astronomical triangle; with the given parts encircled. Since two 
sides and the included angle are given, we may adapt formulas 
(11) to (14) of 155 to the triangle of Fig. 11, or we may con- 




struct an arc of a great circle through M perpendicular to P, V Z, 
letter the triangle as shown in Fig. 12, and then apply Napier's 
rules to obtain 

* If a navigator wishes to observe a number of stars at a particular time, 
say near sunset, he knows the time and from that can find the angle t\ he 
knows approximately what his latitude will be, arid he can find the declina- 
tion of convenient stars in the Nautical Almanac. Hence he can compute 
the approximate positions, altitude, and azimuth of several stars in advance 
and thus expedite the process of locating, identifying, and observing them. 
Instead of computing h and Z> he can find these quantities in tables when 
such are available. 



332 



APPLICATIONS 



[CHAP. XVI 



tan <p = cos t cot d, (3) 

*/ = 90 - L - * = 90 - (L + *>), (4) 

cot Z = cot t sin ^' esc ^> = cot t cos (L + <p) esc ^, (5) 

sin h = cos ^' sec y sin e = sin (L + ^>) sec <p sin rf, (6) 

sin t cos d esc Z sec h = 1 . (Check) (7) 

// L represents the latitude of a place north of the equator, d should 
be taken positive for a body having north declination and negative 
for one having south declination, or vice versa. 

Example. Use formulas (3) to (7) to find the altitude h and 
the azimuth Z n of a star having d = 19'15" S., t = 4510'30" 
east, if it is viewed by an observer in latitude 3730' N. 

Solution. The solution found from the formulas (3), (4), (5), 
(6), and (7) appears below. 



t - 4510'30" E. 
d - -19'15" 
L - 3730'0" N. 
<? = 9138'13" 
L + <p - 1298'13" 
Z - N.1226'43" E. 
h - 339'18" 
1 


(3) 
U cos 9 84810 
Ucot (-)l 09580 

1 tan (-)l 54390 

= Zn 


(5) 
I cot 9 99735 

/ P.SC 00018 
/ cos (-)9 80015 


(0) 
1 sin (-)8 30411 

I hoc (-)l 54414 
/ hin 9 88906 

? sin 9 73791 


(7) 
I sin 9 85080 
/ cos 9 99991 

1 esc 07211 
/ sec 07717 
log 9 '99999 


1 cot (-)9 79708 



Evidently we could have used Napier's analogies to solve the 
triangle of the illustrative example, or we could have adapted 
formula (21) of 156 to the triangle and have used the result to 
find A. 

EXERCISES 

1. From Napier's analogies (148) derive the formulas 



tan 
tan 



- M) = cot \t sin \(L - d) sec i 
+ M) = cot ^ cos i(L - d) esc 



+ d), 
+ d). 



2. From formula (21) of 156, derive the formula* 

hav co-h = hav (L d) + cos 7> cos d hav t. 

* In the practice of navigation the method of Saint Hilairo is frequently 
used to determine the observer's position. In this method the value of 
Z is taken from azimuth tables, and h is computed by the formula of Exer- 
cise 2. The navigator then compares the computed value of h with the 
observed value and uses the difference between the two in determining the 
correction to be applied to the assumed position of his ship. 



167] TO FIND THE TIME AND AMPLITUDE OF SUNRISE 333 
From the data of Exercises 3 to 10, compute h and Z n . 

3. d = 615' S., 7. d = 10 N., 
t = 146' W., < = 40 W., 

L = 2118' N. L = 35 S. 

4. d = 3817'24" S., 8. d = 7 S., 

t = 2830'29" W., / = 28 K, 

L = 2432'58" N. L = 41 N. 

6. d = 5956' N., 9. d -- 8 N., 

= 6032' E., i - 35 E., 

L = 4445' N. L = 39 N. 

6. d = 10 S., 10. d = 2230' S., 

* = 25 E., t = 60 E., 

L = 18 C 57'16" S. L = 45 S. 

From the data of Exercises 11 to 16, compute h. 

11. t = 3 h P.M., 14. < = l h 13 m 12 8 P.M., 
d = 5 S., d = IS^l' N., 

L = 50 N. L = 1554 f S. 

12. = 25 E., 15. t = 4 h 2 m 8" P.M., 
d = 10 S., d = 5956' N., 

L = 1857'i6" S. L = 4445' N. 

13. t = 2 h 40" 1 P.M., 16. t = O h 56 m 24 s P.M., 
d = 10 N., d = 6 15' S., 

L = 35 S. L = 2118' N. 

17. Check the answers of Exercises 3 to 10 using the formulas of 
Exercise 1. 

18. If the observer's latitude is 2917 / 24 // N., and a star, in declina- 
tion 3021'14" S., has the hour angle 4 h 30 48" W., find the altitude 
of the star. Use hav (90 h) = hav (L d) + cos L cos d hav t. 

167. To find the time and amplitude of sunrise. Figure 13 
represents a stereographic projection of the astronomical triangle 
P N ZM when the body M is the sun on the horizon. The dotted 
line indicates the path of the sun across the sky as a small circle 
each of whose points is distant co-rf from the pole. When the sun 
crosses the meridian at K, it is noon. Hence t represents the 
angle through which the earth must turn during the time inter- 
val from sunrise to noon. Since the earth turns through 15 per 
hour, J/15 will be the number of hours from sunrise to noon if t 
is expressed in degrees. The declination of the sun can be found 



334 



APPLICATIONS 



[CHAP. XVI 



from the Nautical Almanac,* and the latitude of the observer is 
supposed known. Therefore, to find a formula for t, apply 
Napier's rules to right spherical triangle NMP N (Fig. 14), and 

write cos (180 t) = tan d tan 

L, or 

cos t = tan d tan L. (8) 

The angular distance from the 
east point of the horizon to 

N 





FIG. 14. 



the sun at sunrise is called the amplitude of sunrise. From 
right spherical triangle NP N M of Fig. 14 we find, by using 
Napier's rules, sin d = cos L sin A, or 



sin A = sin d sec L. 

From Fig. 14 we obtain the check formula 
cot A cot t esc L = 1. 



(9) 
(10) 



Example. Find the amplitude and the time of sunrise at 
AnnapoliSj L = 3859' N., at a time when the declination of the 
sun is 20 S. 

Solution. The solution found from formulas (8), (9), and (10) 
appears below 



L = 3859'0" 
d = -200'0" 
t = 7252'7" 
A = -266'U" 
1 


(8) 
IZtan 9.90811 
[Ztan (-)9.56107 


(9) 
I sec 0.10940 
Zsin (-)9.53405 


(10) 
/ esc 0.20128 

I cos 9.48889 
I cot 0.30983 


/ cos 9.46918 


/sin (-)9.64345 


log 0.00000 



* Owing to refraction of the sunbeams by the earth's atmosphere, the 
sun will appear to be on the horizon considerably earlier than the results 
of this computation would indicate;. In practice, corrections must be made 
on this account. 



168] TO FIND THE TIME OF DAY 335 

Since 15 indicates a time of l h , 7252'7" will indicate 4 h 51 m 28". 
As t is tfye time from sunrise till noon, we obtain 

12 h - (4 h 5l m 28') = 7* 8 m 32' 

as the local apparent time* of sunrise. The negative sign before 
the amplitude indicates that the sun appeared on the horizon 
south of the east point. 

EXERCISES 

1. Find the amplitude of sunrise in latitude 3858'53" N. when the 
declination of the sun is 2229 ; 00" S. 

2. At Annapolis, Lat. 3859' N., the sun in declination 2327' N. has 
the altitude 0. bearing easterly. Find the local apparent time. 

3. Find the amplitude and the local apparent time of sunrise and 
sunset for Annapolis, Md., L = 3858'53" N., at summer and winter 
solstice (d = 23277 // ). 

4. (a) Find the local apparent time of sunrise and sunset at Cape 
Nome, L = 6423' N. on Mar. 21, d = 00'0", Dec. 21, d = 2327' S., 
and June 21, d = 2327' N. (6) Find the amplitude of the sun at each 
occurrence, (c) Find the length of the longest day and of the shortest 
day at Cape Nome. 

5. Assuming that the declination of the sun ranges between 2327 / S. 
to 2327' N., show that a place where the sun rises at midnight must lie 
within 2327' of a pole of the earth. 

Hint. In the formula cos t = -tan L tan d, let / = 180 (= 12 h ). 

6. For a point on the earth having latitude 80 N. find (a) the declina- 
tion of the sun when the time of daylight is just 24 hr.; (6) the declina- 
tion of the sun when the night lasts just 24 hr.; (c) the least altitude and 
the greatest altitude of the sun during the day when the declination 
of the sun is 2327' N.; (d) the declination of the sun when continuous 
night begins; (e) the length of the shortest possible shadow cast by a 
vertical pole 20 ft. long. 

168. To find the time of day. The declination of the sun can 

be found from the Nautical Almanac for a given time, and the 
altitude of the sun can be measured with a sextant. Hence, if 
the latitude of the place is known, the three sides of the astro- 

* The noon of local apparent time occurs when the sun is on the meridian 
of the observer, and the time of day is expressed in terms of the hour angle 
of the sun. Owing to the fact that the sunbeams are refracted by the 
earth's atmosphere, the sun appears to be on the horizon slightly earlier 
than is indicated by the solution given. 



336 



APPLICATIONS 



[CHAP. XVI 




nomical triangle are known, and t can be 
found. Since t represents the angle through 
which the earth must turn before noon if the 
sun is in the eastern sky, and since the earth 
turns through 15 per hour, t/15 will be the 
interval of time before noon if t is expressed 
in degrees. If the sun is in the western sky, 
t/15 is the time since noon. 
To obtain formulas adapted to this caso, substitute from Fig. 15 

a = 90 - A, b = p = (90 - d), c = 90 - L, 

A=t, B = Z, S = %(h + p+L) 
in (22) and (23) of 146, and simplify to obtain 

sin 2 fy = hav t = cos 8 sin (8 h) sec L esc p, (11) 
sin 2 \Z = hav Z = sin (8 - h) sin (8 - L) sec h sec L. (12) 

The law of sines may be used to obtain the check formula 

sin Z esc p esc $ cos A = 1. (13) 

Formula (11) gives the time of day, and formula (12) the angle? 
from which the azimuth Z n of the sun at the time of the observa- 
tion may be determined. 

Example. Find the azimuth Z n of the sun and the local 
apparent time in New York, L = 4043' N., at the instant when 
the altitude of the sun is 3010 / bearing west and its declination 
is 10 N. 

Solution. The solution obtained by using formulas (11), (12), 
and (13) appears below. 



L - 4043' 

h - 3010' 

p = 90 - d - 80 

S - 7526'30" 
S - h = 4516'30" 
S - L - 3443 ; 30" 

t = 5834 ; 9" 
- 3 h 64 m 17" 

Z - N. 10336'20" W. 
Z n - 25623'40" 
1 



(11) 
I sec 12036 

I esc 00665 
I cos 9.40031 
I sin 9.85156 



I hav 9. 37888" 



(12) 

I sec 0.1 2036 
I sec 06320 



I sin 9.85156 
I sin 9. 75560 



I hav 9. 79072* 



(13) 

I cos 9 .93680 
/esc 0.00665 



Zcsc 0.06891 



Zsin 9.98764 



log 0.00000 
* Those who do not use haversine tables may divide log hav t and 



169] ECLIPTIC. EQUINOXES. RIGHT ASCENSION 



337 



Since 5834'9" is equivalent to 3 h 54 m 17" and the sun is in the 
western sky, the time is 3 h 54 m 17" 7. P.M. 

EXERCISES 

1. In formulas (22) and (23) of 146, substitute a = 90 - A, 

fr = p = (90 - d), c = 90 - L, A = t, It = Z, S = \(h + p + L), 
and simplify to obtain formulas (11) and (12). 

2. An observation of the altitude of the sun was made in each of the 
following cities. Find the azimuth of the sun and the local apparent 
time of observation in each case. 

(a) Pensacola, Fla., L = 3021' N., sun's altitude h = 2430' bearing 
east, decimation 2()42 / N. 

(b) Philadelphia, Pa., L = 400 / N., h = 2o0' K., d = 200' N. 

(c) Annapolis, Md., L = 3<)0' N., h = 220 / E., d = 200' N. 
Given the following data, find t and Z. 

3. L = 42 4.ro" N., 6. L = 450 / / ' N., 
d = 18 27'0 /; N., d = 2230'()" N., 
/i = 3836 ; 0" E. A = 30(yO" W. 

4. L = 2535'0" N., 6. 7, = 300'0" N., 
d = 1024'0 // S., d = LWO" N., 
/i = 3519'0" E. A = 450'0" W. 

169. Ecliptic. Equinoxes. 
Right ascension. Sidereal time. 

The earth rotates about its axis 
once a day, and it also moves 
around the sun once a year. 
To an observer on the earth, 
the sun seems to move about the Q'[ 
earth, describing a great circle on 
the celestial sphere called the 
ecliptic. The plane of the eclip- 
tic is inclined at an angle of ap- 
proximately 2327'* to the plane 
of the celestial equator (see Fig. 
16). 

To an observer on the earth the sun appears to move eastward 
on the ecliptic, crossing the celestial equator while moving 

log hav Z by 2 to obtain log sin t/2 and log sin Z/2, respectively, and then 
find t/2 and Z/2 from the table of logarithms of trigonometric functions. 
* This angle 2327' is called the obliquity of the ecliptic. 




338 APPLICATIONS [CHAP. XVI 

northward at the vernal equinox V.E. and while moving south- 
ward at the autumnal equinox A.E. 

The right ascension RA of a body on the celestial sphere is 
the angle measured eastward from the hour circle of the vernal 
equinox to the hour circle of the body; thus the right ascension 
of the sun varies from to 360. Evidently a point is located 
on the celestial sphere by its right ascension and its declination 
just as a point on the earth is located by its longitude and its 
latitude. 

Relative to the stars, the earth turns about its axis once in 
approximately 23 56 m mean solar time. This period of time, 
called the sidereal day,* is divided into 24 equal parts called 
sidereal hours, and the sidereal hours are divided into 60 equal 
sidereal minutes of 60 equal sidereal seconds each. Relative 
to the stars, the earth rotates through 15 each sidereal hour. 
The sidereal time of a place is measured from the time when the 
vernal equinox crosses the meridian of the place. Hence the 
right ascension of the zenith of a place when expressed in hours, 
minutes, and seconds in the usual way is the sidereal time at that 
place. From this it follows that the difference in the sidereal 
times of two points on the earth measures the hour angle between 
their celestial meridians; hence the difference in the sidereal 
times of two points measures the difference in their longitudes. 
A corollary to this may be stated : the difference in sidereal time of 
Greenwich and that of a second place measures the longitude of the 
second place relative to Greenwich as prime meridian. 

Example. At a certain instant the sidereal time at one place 
is 2 , and at a second place it is 4 l 30"\ Find the longitude of 
the second place if that of the first place is (a) 0, (6) 60 E., 
(c) 60 W. 

* Besides sidereal time, we shall consider two other kinds, namely, local 
apparent time and mean solar time. The noon of local apparent time occurs 
when the sun is on the meridian of the observer, and the time of day is 
expressed in terms of the hour angle of the sun. Mean solar time is defined 
in terms of a fictitious sun that travels along the celestial equator at a 
uniform rate and makes a complete circuit in the same time as the actual 
sun. It is mean solar noon when the fictitious sun is on the meridian, and 
the mean solar time at any instant is the hour angle of the fictitious sun. 
This fictitious sun is used in order that we may have a day of uniform length 
throughout the year. 



169] ECLIPTIC. EQUINOXES. RIGHT ASCENSION 



339 




Solution. In Fig. 17 the circle represents the equator. V.E. 
represents the position of the vernal equinox, and A, B, and G 
represent, respectively, the points on the equator where the 
meridian of the first place, that of second place, and that of 
Greenwich meet the celestial 
equator. Since the sidereal time 
of A is 2 h , arc VE A is 2 X 15 
= 30. Similarly, VE B is 67 
and AB = 37^. In case (a), 
Greenwich and A have the same 
meridian; hence the longitude* of 
B is 37^ E. 

In Case ([>), the meridian of 
Greenwich must be represented 
at (?2 in Fig. 17, since A is in 
longitude 60 K. Hence the 
longitude of B in this case i^ 
60 + 37 - 97^ E. 

In Case (r), Greenwich must- have the position (7 3 in Fig. 17, 
since A is 60 west of Greenwich. Hence the longitude of B is 
(>0 - 371 = 22|> W. 

EXERCISES 

1. When it is O l (sidereal time) in (Ireemvirh, it is 4 l at a certain 
place; find the longitude of this place. 

2. At a place in longitude 81lo' W. the sideral time is l() h 17'" 30". 
Find the sidereal time at (ireenwich. 

3. The longitude of a first place differs from that of a second place by 
9530'. When the sidereal time of the first place is 10 , find the sidereal 
time of the second place if it is (a) east of the first place; (6) west of 
the first place. 

4. An observer in longitude 2430' W. observes a star whose RA is 
12 h 31" 1 10". A radio signal gives Greenwich sidereal time at the 
instant of the observation as 4 l 20'" 30". Find the hour angle of the 
star. 

5. If STi is the sidereal time at a first place in longitude Xi west of 
Greenwich and ST* the sidereal time of a second place farther west, 
find the longitude of the second place. 

6. On Jan. 13, 1932, the RA of the star Vega was 18 h 34 m 36". What 
was the hour angle of Vega at the instant when the local sidereal time 
was 12 h 54 m 16'? 



340 



APPLICATIONS 



[CHAP. XVI 



7. At a certain time, the Greenwich hour angle for the Star Rigel 
was 27942' W. Find the local hour angle of Rigel for an observer in 
Long. 7638'30" E. 

170. The time sight. The data and formulas considered in 
168 may be used to find the longitude of an observer whose 
latitude is known. This method of determining longitude at 
sea is called the time sight. In Fig. 18, P N G represents the celestial 
meridian of Green wich, P#0 the celestial meridian of the observer 
and P N M the celestial meridian of the sun. The angle t found 
p by the method of 168 determines 

the local apparent time at O; the 
angle GP*M determines the local ap- 
parent time of Greenwich. Hence 
the longitude in degrees 

X = angle GP N O = angle GP N M - t 

of () is obtained by multiplying by 15 
the difference in hours between the 
local apparent time of Greenwich and 
that of 0. Sometimes it will be neces- 
sary to add angle GP N M and angle t 
and sometimes to subtract them, depending on their relative 
positions. The local apparent time of Greenwich is obtained by 
radio, by telegraph, or by computing it from Greenwich mean 
time shown by a chronometer. The longitude is east or west 
according as the local time is later or earlier than Greenwich 
local time. 

If the object M is a star, we still have 

X = angle GP N M - t, 

where t is computed as in 168, and the angle GPNM is obtained 
by subtracting Greenwich sidereal time (computed from Green- 
wich mean time as given by a chronometer) from the right 
ascension of the star (obtained from a Nautical Almanac). 

EXERCISES 

In each of the following sets of data, ST refers to sidereal time of 
Greenwich, RA to the right ascension of an observed star, d to its 
declination, h to its altitude, and L to the latitude of the observer. 
Find the longitude of the observer for each situation. 




FIG. IS. 



171] 



MERIDIAN ALTITUDE 



341 



1. L = 300'0" N., 
d = 2230'0" N., 
h = 450 / // W., 
*ST = 4 h 10 ra , 
RA = 13 h 5 m . 

2. /, = 120'0" S., 
d = 50'0" N., 
A = 450'0" W., 
8T = 10 h 6 m , 
RA = 8 h 7 m . 

3. L = 390'0" N., 
rf = 200'0" N., 
fc = 220'0" K, 
,ST = 5 h 8 m , 
/M = 2 h () m . 



4. L = 3030'0" N., 
d = 1530'0" N., 
h = 4430'0" W., 
ST = 17 h 15 m 24 8 , 
#A = 10 h 5 m 6 s . 

6. L = 400 / 0' / N., 
d - S^'O" N., 
h = 200 / 0" E., 
^r= O h 47 m 24", 
RA - l h 5 m 7 s . 

6. L = 4330'0 /; N., 
d = IS'W 7 N., 
h = 200 / 0' / W., 
S5T = 13 h 5 in 15 s , 
RA = O h 15 m 20 s . 



171. Meridian altitude. To find the latitude of a place on 
the earth. Figure 19 represents the cross section of the earth 
and of the surrounding celestial sphere 
by the plane of the meridian of an ob- 
server. qq f represents the equator of 
the earth; 2, the position of the observer; 
and PJ>., the axis of the earth. QQ', 
Z, PvPn, N, and S represent, respec- 
tively, the celestial equator, the zenith, 
axis of celestial sphere, north point of 
the horizon, and south point of the hori- 
zon. Since qz represents the latitude of 
the observer and since arc qz = arc QZ 
arc NPN, it appears that the latitude of an observer on the earth is 
equal to the decimation of his zenith and to the altitude of the pole 
elevated above his horizon. 

If, then, an observer knows the declination d of* a star M (see 
Fig. 20) and observes its altitude Af J us ^ as it crosses his meridian 
above the pole, he can find his latitude by writing 

L = NP N = h - (90 - d). 




* The declination of a star can be found from the Nautical Almanac. 
t Various corrections to the observed altitude are generally necessary to 
obtain the true altitude. 



342 



APPLICATIONS 



[CHAP. XVI 



The student should draw a figure for each case. First, a 
figure like Fig. 20 should be drawn showing the circle, Z, N, and 
S. Then the star M should be located on the figure so that 
arc NM = h if the star bears north or so 
that SM = A if it bears south. 

Next, the pole should be located so 
that arc 

AlP N (or MP S ) = 90 - d. 

p s Finally, the altitude of tho pole elevated 
above the horizon should bo computed 
from the figure. 




Fu;. 20. 



Example. Find L if the declination of a star is 62 S. and if 
its altitude as it crosses the meridian at upper culmination* is 
50 bearing south. 

Solution. Since the star bears south and since it appears 
in the sky 50 above the horizon, it is 
represented in Fig. 21 on the right side 
of the circle so that arc SM = 50. 
Next 




MP 8 



is laid off 
latitude is 



90 - d = 90 - 62 = 28 
to locate PS. Hence the 



FIG. 21. 



L = 50 - 28 = 22 S. 



The observer must have been in south latitude since tho south 
pole was elevated above the horizon. 

EXERCISES 

From the meridian altitude A, the declination d, and the bearing of 
the observed body as indicated, find the latitude of the observer in each 
of the following cases: 

* The stars appear to move through the sky, each describing a small 
circle, one of whose poles is the celestial north pole, the other, tho celestial 
south pole. Thus each star crosses the plane of the meridian of a place 
twice every 24 hr., the first time on one side of the pole and the second time 
on the opposite side. The greater of the two altitudes of meridian transit 
is the altitude of upper culmination; the lesser is the altitude of lower 
culmination. 



172] 



GIVEN t, d, h, TO FIND L 



343 



Assume in each of the Exercises 1 to 12 that the body is in upper 
culmination. 



d 

1. 50 N. 

2. 40 S. 

3. 20 N. 

4. 5025' S. 
6. 30 15' S. 
6. 28 10' N. 



h 

40 N. 
20 S. 
60 S. 
3529' S. 
4735 N. 
7112' S. 



d 


h 


1. 4139' N. 


8211 ; N. 


8. 3715 ; N. 


4021' N. 


9. 1 10' N. 


7019' N. 


10. 1739' S. 


7221'S. 


11. 4723' vS. 


352(>' S. 


12. 2313'N. 


7f>4()' S. 



Assume in each of the Kxercises 13 to H> that the body is in lower 
culmination. 

13. 5949' N. 44il' N. 16. 73lo' N. 2848' N. 

14. 7754 ; S. 25 18' S. 16. 4229' N. 2f>23' S. 

17. Two observers, A and Ji, are at different places on the same 
meridian. At the same instan' each observer measured the meridian 
altitude of a star having declination 26 16' S. A observed the star 
bearing south at an altitude 30 17', B observed the star bearing north 
at an altitude 6017'. Find the great-circle distance between .1 and B. 

172. Given t, d> A, to find L. This is the double-solution case, 
since the given parts of the astronomical triangle are two sides 
and the angle opposite one of them. A p 
method of finding L when t, d, and h are 
given is obtained by applying Napier's 
rules to the right triangles in Fig. 22. 
From triangle I, we have cos t = tan y 
tan d or 

tan (p = cos / cot (L (14) 

From triangles I and II, we get 

sin d cos p cos <p, 

sin h = cos p cos </?'. FI. 22. 

Dividing the second of these equations by the first, member by 
member, and solving the result for cos <p', we obtain 

cos <?' = cos (p sin h esc d. (15) 

Then 90 - L = ? + ?', or 

L = 90 - (<p + <?'). (16) 




344 APPLICATIONS [CHAP. XVI 

Two solutions are obtained by choosing <p' from (15), first 
positive and then negative. Since approximate position is 
generally known, only the desired value need be computed. If 
north declination be considered as negative, the latitude found 
from (16) will be north if 90 (^ + #') is positive arid south 
if 90 - (<p + <?') is negative. 

EXERCISES 

1. From the following data, compute in each case the latitude. 

(a) t = 35 W., (6) t = 29 W., 

d = N., d = 7 S., 

h = 42. h = 34. 

2. From the following data, compute in each case the latitude and 
azimuth. 

(a) t = 30 W., (c) t = 3112 / 13 // W., 
d = 15 N., d = 1512'7" N., 

h = 60. h = 5911'44". 

(6) t = 32 W., (d) t = 10 PI, 
d = 26 N., d = 23 S., 

h = 40. h = 22. 

173. MISCELLANEOUS EXERCISES 

1. From cos x = 1 2 hav prove 

sin x sin y hav (x + y) hav (x y), 
cos x cos y = 1 hav (x + ?/) hav (x y), 

and thence, from the law of cosines: 

hav a = hav (6 + c) hav A + hav (b - c) hav (180 - A), 

. _ hav 6 hav (c a) 

hav B = v: -, . r - ? , --- ^ 
hav (c + a) hav (c a) 

or 

- haV (C + a L- 



2. Given t = 4510 / 30" W., d = 19'15" S., L = 3730' N., find the 
azimuth Z n . 

3. Given = 55 E., d = 15 S., and L = 42 N., find /i and Z. 

4. Given t = 30 W., d = 45 N., h = 60, find L and Z. 

5. Given ^ = 30 E., d = 15 S., ft = 60, find L and Z. 



173] MISCELLANEOUS EXERCISES 345 

6. From the following data, compute in each case the latitude and 
azimuth. 

(a) h = 68, (b) t = 30ll f E., 

* = 10 E., d = 2229' N., 

d = 23 S. h = 4457'. 

7. In each of the following exercises, L represents the latitude of the 
observer, d the declination of a star, and h it-* altitude. Find in each 
case the hour angle t and the azimuth Z n of the star. 

(a) L = 45 N., d = 2230' N., h - 30 W. 

(b) L = 30 H., d = 15 N., A = 3730' E. 

8. An airplane following a great-circle track travels from a place 
having L = 375() / N., X = 12220' W. (near Oakland, Calif.) to a 
place having L = 404()' N., X = 7410 / W. (near Newark, N. J.). 
How close does it pass to a point for which L = 4150' N., X = 8740' W. 
(near Chicago, 111.)? 

9. Compute the distance and the intial course for a voyage along a 
great circle from Yokohoma, L = 3526'41" N., X = 13939'0" E., to 
Diamond Head, Hawaii, L = 2151'8" N., X = 15748'44" W. 

10. Compute the distance and the initial course for a voyage along a 
great circle from Brisbane, Australia, L = 2727 / 32" S., X = 1531'48" 
E., to Acapulco, L = lo49'10" N., X = 9955'50" W. Also find the 
latitude and longitude of the southern vertex of the track. 

11. Compute the distance and initial course for a great-circle voyage 
from a point having L = 3742' N., X = 1234' W., near Farallon Island 
Lighthouse, to a point having L = 3450' N., X = 13953' E., near the 
entrance to the Bay of Tokyo. 

12. Find distance and the initial course of a great-circle voyage from 
San Diego, L = 3243 ; N., X = 11710' W., to Cavite, L = 1430' N., 
X = 12055' E. 

13. Find where the track of the preceding exercise crosses the meridian 
of 15749' W. and at what distance from the harbor of Honolulu, L = 
2116 / 5" N., X = 15749' W., then due south. 

14. The initial course by great-circle track from San Francisco, 
L = 3750' N., X = 12230' W., to Yokohama, L = 3530 ; N., X = 
140 E., is 30259'05 // . Find the longitude of the most northerly point 
of this path. 

16. Find the latitude and longitude of the most northerly point 
reached by a ship sailing from San Francisco, Lat. 3748' N., Long. 
122 28' W., to Calcutta, Lat. 2253' N., Long. 8819 ; E. 



346 



APPLICATIONS 



(CHAP. XVI 



16. An airplane follows a great-circle track from New York, L 
4040' N., X = 7410 / W., to /, = 5630' N., X = 30' W. (near Edin- 
burgh, Scotland). Where will it make its nearest approach (a) to the 
North Pole? (b) To L = 4650' N., X = 7110' W. (near Quebec, 
Canada)? 

17. Find the distance in degrees between the sun and the moon 
when their right ascensions are, respectively, 15 h 12 m , 4 h 45 m and their 
respective declinations are 2130' S., 530' N. 

18. Find the distance in degrees between liegulus HA l() h , p 



7719' and Antares RA 



lb' h 2() m , p 



19. An observer in Lat. 6023'20" S. finds the altitude of a star when 
crossing the prime vertical* to be 3823'20", bearing east. Find the 
declination of the star. 

20. A star in declination 4752'15' / S., bearing east, makes its prime- 
vertical transit in altitude 5820'00''. Find the hour angle of the star. 

21. What is the latitude of the place at which the sun rises exactly in 
the northeast on the longest day of the year? 

22. Find the local apparent time of sunrise and sunset at 

(a) London: L = 5129' N., if d of sun = 1317' N. 
(6) Panama: L = 857' N., if d of sun = 1829' N. 

(c) New Orleans: L = 2958' N., if d of sun = 430' N. 

(d) Sydney: L = 3352' S., if d of sun = 430 / N. 

23. Find the length (a) of the longest day; (b) of the shortest day at 
Leningrad L = 5956'30" N., X = 3019 / 22' / K 

24. Find the hour angle and amplitude of moonrise at Washington, 
D. C., L = 3859' N., on a day when the moon's declination is 2528' N. 

26. If twilight continues until the sun is 18 below the horizon, find 
the length of dawn, dark night, bright day, and twilight in Annapolis, 
L = 3858'53" N. (a) at summer solstice (d = 2327'7 / ' N.); (b) winter 
solstice (d 2327 / 7 // S.); (c) when the sun is at an equinox. 

26. The following observations have been made of a heavenly body 
in upper culmination. Find the latitude in each case. 





Declination 


Observed altitude 


Bearing 


(a) 


2810' N. 


7112 / 


South 


(W 


7302' N. 


SS^O' 


North 


(c) 


4417' S. 


6523 ; 


South 


(d) 


3015' 6. 


4735' 


North 


W 


5025 ; S. 


3529' 


South 


(/) 


4016' N. 


4014 / 


North 



* For definition of prime vertical, see 164. 



173] MISCELLANEOUS EXERCISES 347 

27. What relations must exist between L and d for a lower culmina- 
tion to be visible? What relation always exists at a visible lower 
culmination between h and d? 

28. In each of the following observations of a lower culmination, find 
the latitude: 





Declination 


Observed altitude | 

I 


Bearing 


(a) 
(*>) 
(c) 
(d) 


8850' X. 
4622' S. 
5949' X. 
7754' S. 


3720' ; 
32 W 1.V | 

44 o ir ! 

2518' i 

1 


Xorth 
South 
Xorth 
South 



29. The right ascension of the sun is 45; find (a) the length of the 
night at a point in latitude (>0 X.; (b) the length of the shadow cast by 
a vertical stick 10 ft. long at 10 A.M. (local apparent time) at a point in 
latitude 40 X.; (c) the direction of a wull tlmt casts no shallow at 
10 A.M. at a place having latitude 40 X. 

Hint. Compute the declination of the sun iind then dra\\ the 
astronomical triangle. 

30. At a place in Lat. f>l32' X., the altitude of the sun is 3;>1,V 
bearing west and its declination is 2127' X. Kind the local apparent 
time. 

31. In London, L . r >l3l' X., for an afternoon observation the alti- 
tude of the sun is 1,">40'. If its declination is 12 S., iind the local 
apparent time. 

32. (a) A navigator in latitude lf>23'30" S. observes a star having 
RA = 12 h 27 32", d = 2210'3ti" N., at an altitude h = 172'30" W. 
If the sidereal time ST of Greenwich at the instant of observation is 
10 b 27'" 34", find the longitude of the navigator. 

(b) Also find the longitude of a second navigator in latitude 
6221'39" X. who at the same instant observes a star having RA = 
( ,h 27 n. ;3Q ^ d = 2o55'21 // N. at an altitude h = 3317'44" W. 

33. Find to the nearest minute the direction of the shadow of a ver- 
tical staff in Lat. 3850' N. at (5 A.M. local apparent time, when the 
declination of the sun is 2327 ; N. 

34. Find the direction of a wall in Lat. 5230' N. that casts no shadow 
at 6 A.M. on the longest day of the year. 

35. An explorer claimed to have reached the north pole. He took the 
picture of a flagpole 6 ft. high. From measurements made on the 
photograph it appeared that the 6-ft. pole cast a shadow 10.1 ft. long. 
Prove that he must have been at least 7 from the pole. 



348 



APPLICATIONS 



[CHAP. XVI 



Find the shortest length of shadow that a stick 6 ft. long could pos- 
sibly cast on level ground when held vertical at the north pole. 

36. If the altitude of the north pole is 45 and if the azimuth of a 
star on the horizon is 135, find the polar distance of the star. 

37. Find the time of day when the sun bears due east and when it bears 
due west on the longest day of the year at Leningrad (Lat. 5956' N.). 

38. Two points on the earth are in latitude 40 N. and their differ- 
ence in longitude DLo = 70. How much does the parallel of latitude 
joining these points exceed in length the arc of the great circle joining 
them? How far apart are the mid-points of the two tracks? (Use 
3437 nautical miles for the radius of the earth.) 

39. Find the altitude of the sun at 6 h A.M. at Munich (Lat. 489' N.) 
on the longest day of the year. 




40.* If a, 6, c, and A refer to a spherical triangle and if in Fig. 23 
OY = hav (6 - c), MZ = hav (b + c), OE = hav A, and OM = 1 
unit, prove that x(=. EX MR) is equal to hav a. 

* This plan was devised by Prof. John Tyler, U. S. Naval Academy. 



173] MISCELLANEOUS EXERCISES 349 

Hence, if we take OP = OM = MN = 1 unit, make a scale on OP by 
marking angles 6 between and 180 at points on OP distant in each 
case hav 6 from 0, a scale on OM by marking angles 8 at points on OM 
distant in each case hav from 0, and a scale on MN by marking 
angles 8 at points on MN distant in each case hav from M, show how 
we may find the third side of a spherical triangle, when two sides and 
the included angle are given, by drawing three straight lines and 
reading the result. 



APPENDIX A 

1. The mil. The mil is an angular unit equal to ^iW f f ur 
right angles. 

The word mil, meaning one-thousandth, originated from the 
idea of adopting as a unit the angle that subtends an arc equal 
to i'oVo f t*h e radius. Such an angle subtends 1 ft. at a distance 
of 1000 ft., 1 yd. at a distance of 1000 yd., etc. This manifestly 
furnishes a quick method of estimating the distance of an object 
whose size is known. There would under these circumstances 

be f\7\ryT or 6283.18+ such units subtended by a circle. This 

number is too inconvenient to be of practical use in calibrating 
instruments. The circle is therefore divided into 6400 equal 
parts, and each of these is called a mil. The arc subtended by a 



central angle of 1 mil therefore equals Tnjr or (0.00098 + )/, or 



so nearly xtf^ of the radius that it may be so taken for purposes 
not demanding great accuracy. This property, coupled with the 
knowledge that in small angles the chord very nearly equals the 
arc, enables us to say for rapid and rough approximation : 

A mil subtends a chord equal to -f^oTF f ^ e distance to the chord. 

With due regard to the degree of approximation, a small number 
of mils (several hundred) subtends a chord equal to the small number 
times iQoo of the distance to the chord, or, in symbols 







1000 



where 6 is in mils and s and r are expressed in the same unit. 

The methods of rapid approximate measurement of angles and 
distances by the use of the mil system were first developed by the 
Field Artillery in computing firing data. Their use was extended 
to mapping, sketching, and reconnaissance. During the World 
War the Infantry adopted the system, and it has now become 
general. 

351 



352 PLANE AND SPHERICAL TRIGONOMETRY 

The mil as a unit has the advantage that it is convenient in 
size for certain military measurements. 

Example 1. Two points, A and B, are 50 yd. apart and 
2000 yd. away. How many mils should they subtend (see 
Fig. 1)? 




Observer 2000 yd 

FIG. 1. 

Solution. 50 divided by f-JHHr = 25. 

Or, at 2000 yd., 2 yd. corresponds to 1 mil; therefore 50 yd. 
corresponds to 25 mils. 

Example 2. An observer measures the angular distance 
between two points, A and B, 5000 yd. away, to be 30 mils. 
How far apart are A and B! 

Solution. %%%% X 30 = 150. 

Or, at 5000 yd., 1 mil subtends 5 yd.; therefore 30 mils sub- 
tends 150 yd. 

Example 3. The angular distance between A and B is observed 
to be 40 mils. They are 100 yd. apart. How far away are they? 

Solution. ^o X 1000 = 2500. 

Or 40 mils corresponds to 100 yd.; therefore 1 mil corresponds 
to 2 yd., but 2^ is y^^ of 2500 yd. 

EXERCISES 

1. A battery with a front of 60 m. is observed from a point 3000 m. 
away, measured on a line normal to the battery. What angle does the 
battery subtend? (Or what is its front in mils?) 

2. A four-gun battery 4000 m. away has a front of 15 mils. How 
many meters between muzzles? 

3. The guns in your battery have wheels 1^ m. in diameter. You 
measure a wheel as 5 mils. How far arc you from the battery? 

4. An observer measures the front of a target to be 40 mils at a 
point 6000 m. away. What should a scout (a) 3000 m. in front of the 
same observer measure it to be? (6) 4000 m. in front of the observer? 



APPENDIX A 353 

5. Two targets, T and , are 20 m. apart. The range TO, perpen- 
dicular to the line of targets, is 5000 m. Two guns, G and g, are also 
20 m. apart, the angle TGg being 1500 mils. Take t and g both on the 
same side of TG. 

(a) What is angle tgG in order that the gun g may be laid on t? 

(b) What change in deflection of G must be given to lay it on t! 

6. A hostile trench measures 80 mils from your position. A scout 
500 meters in front of you measures it 100 mils. What is the distance 
of the trench from your position? 

7. You signal to a man at a distant tree to post himself 20 yd. 
from the tree (measured perpendicular to the line from the tree to 
you). The man is now 8 mils from the tree. How far away is the 
tree? 

8. An observer finds that he is on the same level with the top of a 
distant tower that is 34 yd. high. The angular depression of the base 
of the tower is 8 mils. How far away is the tower? 

9. From D a distant object B appears to the right of an object A, 
which is 6000 meters away. An observer at D measures the angle 
ADB to be 35 mils. He moves to C, 180 meters to the right on a line 
normal to AD, and measures the angle A CB to be 15 mils. How far 
away is #? f 

Hint. Sum of angles of a triangle is constant. 

10. From Trophy Point, near the U. S. Military Academy, the 
angular elevation of Fort Putnam is 210 mils, and its distance is 600 yd. 
Also, the elevation of the top of the West Academic Building is 120 mils, 
and its distance is 250 yd. The West Academic Building and Fort 
Putnam are 500 yd. apart. What is the angular elevation of Fort 
Putnam as measured from the top of the West Academic Building? 



APPENDIX B 



2. The range finder. A range finder is an instrument designed 
to obtain the distance of an object from the instrument. Essen- 
tially it is a mechanism in a tube by means 
of which images caught at the ends of the tube 
can be brought into alignment by turning a 
thumbscrew. 

In Fig. 2 line AB represents a range finder 
of length b. AC and BE are lines perpendic- 
ular to AB. When the two images of point 
C caught at the ends A and B are brought 
into alignment, the distance AC = R can bo 
read on a dial. When the image of point C 
caught at end A is brought into alignment 
witl the imago of point D caught at B, the 
distance AG = R\ is registered on the dial. 

The distances R and R\ in Fig. 2 must be so 
great as compared with b that the errors in 
the equations 




R<t> = 6, 
b 



Ri8 = &, 
b 



(1) 

(2) 



are negligible. On the other hand when the range of an object is 
so great that the angles represented by and 6 in Fig. 2 are small, 
relative to the errors inherent in the mechanism of the range* 
finder, trustworthy results cannot be obtained. A 12-ft. range 
finder is effective for distances from 100 to 25,000 yd.; a 26-ft. 
instrument, for ranges from 1200 to 50,000 yd.; a 30-ft. instru- 
ment, from 2400 to 60,000 yd. 

The following examples illustrate the? principles involved in the 
use of range finders. 

Example 1. Let Fig. 2 represent a range finder of length 
6 set parallel to line CD. If 6 = 10 yd. and if the distance 

354 



APPENDIX B 



355 



RI = 2500 yd. and R = 10,000 yd. have been found by using 
the instrument, find the length of CD. Also find CD in terms of 
R, RI and b. 

Solution. Denote angle EEC by <t> and angle EBD by 0. 
Since these angles are small, use equations (2) to obtain 

i__u_, A_ 10 . 

jfg ~~ 10000' R\ ~~~ "2^6 6 
By using (I), we obtain 

CD = RO /</> = 1()000[^$Q T7f(fo(rl = 30 yd. (approx.). 
To find CD in terms of R, R^ and 6, use (2) and (1) to obtain 

<t> = jj* 6 = -^-^ CD = 7(0 0), (approx.). 



Replacing (0 0) in the last equation by their values from the 
first two, we obtain 



-Jt( b - b \- 
~ K \R, R) ~ 



(3) 



Example 2. Figure 3 indicates how a range finder may be 
used to obtain the direction angle a for an D F E 

object CD of small known length a by 
means of the ranges R and RI which may be 
read from the instrument. Find angle a in 
terms a, 6, R, and i?i, assuming that a and 6 
are small as compared with R and RI. Find 
a if a = 50 yd., R = 3000 yd., RI = 1000 
yd., and b = 10 yd. 

Solution. Referring to Fig. 3, observing 
that CF is small and using (3) in the solution 
of Example 1, we have 



p D 



(approx .). 



Since angle FCD = a, sin a = sin (FCD) = 
FD/a, or replacing FD by the value just found, 




sin a 



b(R - JtQ 



(4) 



356 PLANE AND SPHERICAL TRIGONOMETRY 

For the values mentioned in the example, 

10(3000 - 1000) 
-" - 



"50(3000) 



. 
and 



Example 3. A range finder is poorly adjusted. Show how 
the range given by such an instrument may be corrected. 

Solution. When a range finder is not well adjusted it will 
register inaccurate distances. Referring to Fig. 4, we may say 
D a C E in such a case, that the ranges R and RI 
*" are based on angles <t> d and d where 
d is the error due to poor adjustment of the 
instrument. Hence 




<* - 



d = (6) 



If x is the corrected range, we have x 
(d 0) = a, since d and <f> are the true 
angles. Then we may write 



A B 

FIG. 4. 



X 



8 - 



(0 d) - (<t> d) 



(6) 



or, replacing dby b/Ri and <t> d by ^ from (5), we obtain 
the corrected range 



x = 



b_ 
R 



b(R - 



(7) 
2100 yd., 



For example, if a = 50 yd., R = 12,000 yd., 
and 6 = 10 yd., the corrected range would be 

50(12,000) (2100) _ 
X 10(12,000 - 2100) ~ 12 ' 727 yd -' 

and the correction increment is 727 yd. 

EXERCISES 

1. In Fig. 2 find (a) CD if R = 10,000 yd., Ri = 2000 yd., and b = 
30 ft. (b) R if fli = 1500 yd., CD = 180 ft., 6 = 36 ft. (c) CD if 
= 990", = 165", b = 36 ft. 



APPENDIX B 



357 



2. In Fig. 3 find (a) a if R = 10,000 yd., Ri = 2500 yd., a = 180 ft., 
b = 36 ft. (b) find a if <t> = 188", = 960", a = 165 ft., 6 = 30 ft. 
(c) find a if a = 930', 7^ = 3500 yd., Ri = 1000 yd., 6 = 30 ft. 

3. In Fig. 4 find the correction increment (a) if R = 15,000 yd., 
R, = 2800 yd., CD = 165 ft., 6 = 36 ft. (6) if * = 185'', = 545", 
b = 48 ft., CD = 300 ft. 



4. In Fig. 5 6 = 36 ft., (a) find DT if R = 
14,000 yd., Ri = 2000 yd., R* = 800 yd. (b) 
find DT and DC if a = 70, < - 155'', 0, = 
17 10", 0. 2 = 4200". 




FIG. 5. 

5. The captain of a vessel equipped with a coincident range finder of 
effective length 30 ft. desires to find the distance between two channel 
buoys C and D. He trains his range finder on buoy C and reads range 
R f = 14,000 yd. He then aligns the image of D with the image of (' 
and reads on the dial RI = 2000 yd. If the range finder is parallel 
to CD for the readings, find the distance between the buoys. 

6. Two masts on a freighter are 165 ft. apart. The captain of a 
cruiser wishes to find the distance to the freighter with a range finder 
that is poorly adjusted. He trains the range finder on the right-hand 
mast and reads on the dial 15,000 yd. He then aligns the image of the 
second mast with that of the first and reads on the dial 2800 yd. If 
the range finder is parallel to the freighter, find the corrected range 
and the angular error of for his instrument. 



APPENDIX C 



3. Stereographic projections. In the applications of this 
chapter, the student will frequently find it convenient to draw a 
figure showing the main features of the problem under considera- 
tion. For this reason the following facts relating to stereographic 
projections are presented. 

Consider a plane through the center of the sphere in Fig. 6 
and the poles P n and P 8 of the great circle in which the plane 
intersects the sphere. A straight line connecting any point P 
on the sphere to P s cuts the plane in a point called the stereo- 
graphic projection of the point. The stereographic projection of 
a curve lying on the sphere is the locus of the stereographic 

projections of its points. The 
point P 8 is called the center of 
projection, the plane is called the 
primitive plane, and the great 
circle cut out by the primitive 
plane is called the primitive circle. 
The angular measure of an arc 
of a great circle that has a given 
arc as a projection is called the 
true length of the given arc. 

Figure 6 represents the sphere 
with center of projection P a , 
with primitive plane WSEN, 
and with p the stereographic 
projection of P. The truth of the following statements, num- 
bered I, II, III, IV, and V, is easily perceived. 

I. The points of the hemisphere on the same side of the 
primitive plane as P 8 project outside the primitive circle, and the 
points on the other hemisphere project inside the primitive circle. 

II. The projection of any great circle through the center of 
projection P 8 is a straight line through the center of the primitive 
circle. 

III. The primitive circle projects into itself. 

358 




APPENDIX C 



359 



IV. The projection of any great circle passes through the ends 
of a diameter of the primitive circle. For the plane of the great 
circle cuts the primitive circle in a diameter and the ends of this 
diameter project into themselves. 

V. The part of the projection of an arc, of a great circle that 
lies inside the primitive circle has a true length of 180, and if 
this arc is bisected each part has a true length of 90. 

The following statements, numbered VI and VII, are of 
fundamental importance. The proofs are omitted. 

VI. The stereographic projection of a circle lying on a sphere 
is a circle or a straight lino. 

VII. The angle of intersection of two arcs on a sphere is equal 
to the angle of intersection of their stereographic projections. 

4. Construction of some simple projections. The projection 
of a great circle can be drawn when the two points where it 





FIG. 7. 



FIG. 8. 



(Tosses the primitive circle at the ends of a diameter and the 
projection of another point are known. For, by VI, 3, the 
projection is a circle three points of which are known. For 
example, suppose that a great circle cuts the primitive circle 
shown in Fig. 7 at point M \ and that A is the projection of 
another of its points. If is the center of the primitive circle, 
MI lies on the projection by IV, 3. Therefore the circle through 
MI, A, and M 2 is the required projection. Only the stereographic 
projection of one-half of a great circle is shown in Fig. 7. 

Again, the projection of a great circle can be drawn when a 
point where the great circle cuts the primitive circle and the 
inclination of the plane of the circle to the primitive plane are 



360 PLANE AND SPHERICAL TRIGONOMETRY 

known. For, by IV, 3, two points at the ends of a diameter are 
known, by VI the projection is a circle, and by VII the angle 
between the primitive circle and the projection arc known. 

Suppose that the great circle whose stereographic projection 
is to be drawn cuts the primitive circle GM\K shown in Fig. 8, 
at Mi and that its plane is inclined 35 to the primitive plane. 
Draw the mutually perpendicular diameters MiM z and GK, 
construct with a protractor the line MiT, making an angle of 35 
with OM i and meeting GK at S. With S as a center and SMi 
as radius, draw the required circle MiRM z . The circle sym- 
metrical over MiMz with the one drawn also satisfies the given 
conditions. 

EXERCISES 

1. What great circles project into straight lines? 

2. What is the nature of the projection of any circle passing through 
the center of projection? 

3. What is the true length of the arc M\R in Fig. 3? Give a reason 
for your answer. 

4. Construct the projections of the great circles whose planes are 
inclined at 30, 60, 90, 120, and 150, respectively, with the primitive 
plane, assuming that each one passes through a point M\ chosen on the 
circumference of the primitive circle. 

6. Draw a circle to be used as primitive circle. Through the ends of 
one of its diameters construct a circle. This second circle is the pro- 
jection of a great circle. Now construct the projections of two other 
great circles through the ends of the same diameter, each of whose 
planes is inclined at 30 to the plane of the great circle whose projec- 
tion is drawn first. 

6. To find the true length of a projected arc. The actual 
magnitude of an arc of a great circle that has a given arc as its 
projection has been called the true length of the given arc. The 
object of this article is to give, without proof, a method of finding 
the true length of any arc that is the stereographic projection 
of a part of a great circle. 

Let arc ACB in Fig. 9 represent the projection of a great circle 
on the primitive plane ABF. It passes through the ends A and B 
of a diameter and cuts the perpendicular diameter EF at C. 
Draw line AC and prolong it to meet the primitive circle in D, 



APPENDIX C 



361 



lay off arc DG equal to 90 toward the inside of the projected 
circle, and draw GA meeting EF at X. The true length of 
arc ST is then obtained by drawing XS and XT to meet the 




FIG. 9. 

primitive circle in Si and Ti, respectively, and then using a 
protractor to find the length in degrees of arc SiTi. 

If the method just described be applied to find the true length 
of a part of a diameter, the point X, will be found to fall at the 
ond of the perpendicular diam- 
eter. Hence, the true length of 
OC in Fig. 9 is the arc BD, and 
the true length of XC is the arc 
GD or 90. It now appears 
that X is the projected pole of 
the great circle represented by 
ACB in Fig. 9; consequently 
we may refer to X as the pole 
of great circle ACB. 

Evidently we can now lay off 
an arc of any desired true 
length from a given point on a 
projection of a great circle. 
Thus, to lay off 50 from A 
along the arc ACB in Fig. 10, lay off arc AT equal to 50, locate 
the pole X of arc ACB, and draw XT meeting arc ACB in E. 
The arc AE has a true length of 50. 




D 



362 PLANE AND SPHERICAL TRIGONOMETRY 

Note that arc AC = 90, and arc AO = 90. Therefore, in 
accordance with a theorem from solid geometry, angle OA C is 
measured by the true length of arc CO, or by arc DB. A little 
reflection on the processes just illustrated will enable the drafts- 
man to measure with facility angles and arcs defined by projec- 
tions of great circles. 

To measure the angle between two projected arcs of great circles 
through point A, lay off arc AD = 90 on one circle and arc 
AE = 90 on the other, draw straight lines AD and AE to meet the 
primitive circle in D and E, respectively, and measure arc DE with 
a protractor. Since A is the pole of arc DE and angle A is meas- 
ured by the true length of arc DE, the reason for the construction 
is apparent. 

Also, the angle between two arcs may be obtained by measuring 
the angle between their radii drawn to the point of intersection. 

EXERCISES 

1. Draw a primitive circle and the projections of three great circles 
making 45, 90, and 135 angles, respectively, with the primitive and 
ail passing through the ends of the same diameter. Divide each arc 
inside the primitive circle into six parts, each having a true length of 
30. Also check the angle between the primitive and the projection by 
finding the true lengths of jtarts of the diameter perpendicular to the 
one having its end on the projected circle. 

2. Draw the projections of two great circles meeting in a point A 
inside the primitive circle. Lay off arc AD = 90 on one projection 
and arc AE = 90 on the other. Now find the true length of arc KD\ 
that is, measure the angle EAD. Perform this operation three or four 
times, using different great circles in each case. 

3. Through the ends A and li of the diameter of a primitive circle 
draw a projected circle making a 00 angle with the primitive circle. 
Lay off arc AC equal to 60 on the primitive circle and draw through 
the ends C and D of a diameter the projection of a great circle making a 
45 angle with the primitive. Now measure all arcs and angles formed 
inside the primitive circle. 

6. To measure the parts of a spherical triangle by stereographic 
projection. A spherical triangle can be solved graphically by 
drawing its projection and measuring its sides and angles. An 
example will illustrate the method. 



APPENDIX C 



363 



Example. Use stercographic projection to solve the triangle 
in which side b = 120, side c = 75, and the included angle 
A = 60. 

Solution. The solution will be explained by referring to 
Fig. 11. Draw the primitive circle ACF. Then draw any 
diameter AE and the perpendicular diameter DF. Lay off 
arc ADC = b = 120. Draw AO^ so that angle OAOi = 60. 
With OL as center, draw circular arc ABE. Then angle DAB = 




Fio. 11. 

60. Find the pole Xi of arc ABE, lay off arc ABi = 75, draw 
BiXiio meet arc ABE in B. Then arc AB has a true length of 
75. Now draw diameter CG and construct the circular arc 
CBG with center 2 . Then triangle ABC is a stercographic 
projection of the required triangle. To measure the unknown 
parts, draw diameter LM perpendicular to CG, and locate the 
pole X 2 of arc CBG. Draw X 2 B to meet the primitive circle in 
B 2 . Then the true length of CB is equal to arc CJS 2 , which if- 
found by means of a protractor to be 74. Next draw 2 C. 
Then angle BCD is equal to angle GC0 2 = 5830'. Also, angle 
CBA is 180 - angle OiJ50 2 or 



364 PLANE AND SPHERICAL TRIGONOMETRY 

i 

EXERCISES 

1. Draw the stereographic projection of a spherical triangle in 
which a = 60, b = 90, C = 60, and measure B and c. 

2. Draw a stereographic projection of each of the spherical triangles 
that have the given parts indicated, and measure the unknown parts: 

(a) a = 60, (c) A = 120, 

b = 60, b = 75, 

C = 90. c = 150. 

(6) A = 60, (d) b = 120, 

B = 60, c = 120, 

c = 120. A = 75. 



INDEX 



Abscissa, 50 

Accuracy, of computation, 18, 67, 
248 

of five-place tables, 68, 79 
Addition formulas, 120, 125 
Altitude, 331 

Ambiguous case, plane triangle, 
156 

right spherical triangles, 281 
Amplitude, 114, 203, 334 
Angle, 48 

defined as rotation of ray, 48 

of depression, 72 

of elevation, 72 

subtended by a line, 72 
Antilogarithm, 226 
Arc length, 1)6 

Area, of oblique triangles, 155, 
164, 169, 172 

of regular polygons, 93 

of right triangles, 27 
Astronomical triangle, 330 
Axes, 50 
Axis, of earth, 322 

of imaginaries, 202 

of reals, 202 
Azimuth, 331 



Celestial equator, 328 
Celestial meridian, 329 
Celestial pole, 328 
Celestial sphere, 328 



Central angle of a circle, 96 
Characteristic, 221 

rules for determining, 222, 223 
Circle, great, 269 

unit, 106 

vertical, 330 

Circular arc, length of, 96 
Clausen's formula, 188 
Co-function, 29 
Cologarithm, 229 
Complex numbers, 201 

conjugate, 201 

exponential forms of, 211 

geometrical representation of, 
202 

multiplication of, 201, 205 

operations involving, 201 

polar form of, 203 

powers and roots of, 208 

pure imaginary, 200 

quotient of, 201, 206 
Computation, suggestion for, 231 
Conversion formulas, 133, 134 
Coordinates, rectangular, 49 
Cosecant, defined, 10, 52 
Cosine, defined, 6, 52 
Cosines, law of, for plane triangles, 
146 

for spherical triangles, angles of, 

293 

sides of, 291 

Cotangent, defined, 10, 52 
Course, 73, 323 
Coversed sine, defined, 10 
Culmination, 342 



365 



366 



PLANE AND SPHERICAL TRIGONOMETRY 



Declination, 330 
Degree, 48, 265 
Delambre's analogies, 302 
DeMoivre's theorem, 208 
Departure, 73 
Depression, angle of, 72 
Difference in latitude, 73 
Distance, 53, 73, 323 
Double-angle formulas, 120 
Double solution, spherical triangle, 
304, 343 

E 

Ecliptic, 337 

Elements of a triangle, 70 
Elevation, angle of, 72 
Equator, 322 
Equinoctial, 328 
Equinox, autumnal, 338 

vernal, 338 
Euler, Leonard, 212 
Exponents, laws of, 216 



Folded scales, 247 

Forms, directions for making, 230 

used in the solution of oblique 

triangles, 153 
Formulas, addition, 120, 125 

conversion, 133, 134 

double angles, 129 

for half-angles, 129, 164 

of spherical triangles, 295 
Function, 5 
Functions, hyperbolic, 212 

inverse, 177 

of k 90 0, 104 

natural, 16, 69 

of 90 - 0, 99 



Functions, of 90 + 0, 270 + 0, 

180 0, -0, 101 
reciprocal, 9, 28 



Gauss' equations, 302 

Grade, 97 

Graphs, of inverse functions, 17S, 

179 

of Iog 10 x, 237 
of trigonometric functions, !()(> 

to 114 
Greenwich, meridian of, 323 

H 

Half-angle formulas, 129. 104 
of spherical trigonometry, 295 

Half line, 48 

Haversine, dc lined, 10 

Haversine solution, 315 

Heron's formula, 169 

Horizon, 329 
points of, 329 

Hour angle, 330 

Hour circle, 329 

Hyperbolic functions, 212 



I 



Identity, defined, 30 

fundamental, 28, 29, 32, 58 

method of verifying, 34 
Imaginary numbers, 200 
Interpolation, 225 
Inverse functions, 177 

equations involving, 185, 195 

general value of, 180 

graphs of, 178, 179 

principal value of, 182 

relations among, 184 



INDEX 



36 7 



Latitude, 322 

to find, 343 

Length of circular arc, 96 
Line of sight, 72 
Line values, 40 
Logari'i .^s, autilogarithms, 226 

chan^r of base in, 235 

characteristic of, 221, 222, 223 

common, 221 

in computation, 227, 230, 231 

defined, 217 

forms foi , l. r <3 

laws of, 218 

mantissa of, 221, 224 
Longitude, 323 

M 

Machines formula, 188 
Magnitude, order of, 290, 305 
Mantissa, 221, 224 
Meridian, 322 
prime, 323 

Meridian altitude, 341 
Mil, 97, 351 
Minute, 48 
Modulus, 203 
Mollweide's equations, 144 

N 

Nadir, 329 

Napier's analogies, 300 
Napier's rules, 274 
Nautical mile, 96 



Oblique spherical triangles, 269 
Oblique triangles, 152 
Order of magnitude, 290, 305 
Ordinate, 50 



Parts, circular, 274 

of triangle, 70 
Period, 113 
Polp.r form, 203 
Polar triangles, 282 
Pole, North, 322 

South, 322 
Prime meridian, 323 
Prime vertical, 330 
Primitive circle, 358 
Primitive plane, 358 
Principal values, 182 
Proportion principle, 241, 249 
Proportional parts, 225 
Pure imaginary numbers, 200 
Pythagorean theorem, 32 

relations derived from, 32, 38 

Q 

Quadrantal triangles, 284 
Quadrants, 50 
rule for, 277, 278 

R 

Radian, 94, 265 

Range finder, 354 

Ratio, 3, 4, 5 

Ray, 48 

Reciprocal functions, 9, 28 

Rectilinear figures, 20, 40, 42, 85 

Right ascension, 338 

Right spherical triangles, 271 

formulas relating to, 271 

Napier's rules for, 274 

solution of, 278 

Right triangles, directions for solv- 
ing, 71 
Roots, cube, 255 

square, 253 

Rotation, positive and negative, 
48 



368 



PLANE AND SPHERICAL TRIGONOMETRY 



S 



Saint Hilaire, method of, 332 

Secant, denned, 10, 52 

Second, defined, 48 

Sidereal time, 338 

Sine, defined, 6, 52 

Sines, law of, for plane triangles, 

140, 260 

for spherical triangles, 289 
Slide rule, 241 

body, 243 

hairline, 244 

indicator, 244 

scale, 241 

slide, 243 



Triangle, area of, 27, 155, 164, 169, 

172 

astronomical, 330 
elements of, 70 
oblique, 152 
plane, cases of, 152, 153, 156, 

161, 167 
right, 17, 70 
right spherical, 271, 278 
solution of, 70, 76, 78, 82 
by slide rule, 259-265 
spherical, 269 

cases of, 295, 298, 303, 304, 

309, 312, 315, 316, 319 
terrestrial, 323 
Trigonometric equations, 189, 192 



Solid geometry, propositions from, Trigonometric functions, defined 



270 

Spherical triangles, 269 
Stereographic projections, 358 
Subtraction formulas, 122, 125 



Tables, of common logarithms, 
224 

of natural functions, 15, 16, 68, 
69 

of trigonometric functions, 79 
Tangent, defined, 5, 52 
Tangents, law of, 144 
Terrestrial triangle, 323 
Time, local apparent, 338 

mean solar, 338 

sidereal, 338 
Time sight, 340 
Transit, 342 



as ratios, 6, 10 
definitions of, 6, 10, 52 
expressed in terms of one, 38 
of 45, 30, 60, 0, 90, 12 
inverse, 177 

table of values of, 16, 69, 79 
Tyler, John, 348 



U 



Unit circle, 106 



Versed sine, 10 



Z 



Zenith, 329 



ANSWERS 

3. Pages 8, 9 

343434 3 l 1 10 1 

2. |, J. f J jh f, ; -7=. --, -r. 3; - - = .. -7=, A 

v 10 v 10 vioi VIM 

3. cos /I = Y i tan A = A 6. cos A = -}-y, tan A = A 

4. sin A = f , tan A = 2 7 4 7. sin A = T^, tan A = ^ 7 T 
6. sin A = /y, cos A = YT 8. sin A = ^, tan A = ^7- 

11. 550 ft. 13. 9 ft. 16. 1500 ft. 

12. 1120ft. 14. 198.5 ft. 

4. Pages 11, 12 

Q C -I -| 

I- , - / -' %, etc- ; f , $, :], ( t.; |, ^, f , etc.; , - - , 1, etc.; 
V34 V 34 . V2 V2 

1 9 

- 1 * 1 ,,. .21 20 21 *,. 

7-' ~^' ^> L^L. y 29^, ^9, ^Q, etc'. 



2. A, If, A, ^c.; H, A, -V 1 . etc. 

3. (a) cos = 4, tan = 4> etc.; (6) sin = A T? etc.; (c) sin = 

2 

tan = x/3, etc. 

4. (a) 1; (6) 1 7. 45.0ft. 
6. 180 ft. 8. 396 ft. 

6. Pages 14, 16 

2. 0.000291, 1, 0.000291, etc.; 1, 0.000291, 3436, etc. 

3. 0, 1, 0, etc.; 1,0, , otc. 

4 - A, ir A, etc -J TT, TT - 4 1T, etc. 

6. ^ *, A/3, etc. 6. -L, -^, 1, etc. 



\/2 

9. (a) -^= (6) \/6, (c) 1, (d) ~- 13. 0.577 miles 

3\/2 



11. ~ - ?-= ^, etc. 14. 22.5 ft. 

V13 



12. 1 >-(\/3 + 1), - ^ - 16. 482.8 yd 

2V2 2(V3 - 1) 

7. Pages 18 to 20 

1.0- 41.80, 6 - 49.79; 6 - 62.92, c = 97.88; a = 140.8, c - 812; 
a - 96.14, c - 102.3 

369 



370 



PLANK AND SPHERICAL TRIGONOMETRY 



2. (a) a - 48.79, b - 69.62; 

(b) b - 1134, c - 1152; 

(c) 350, 610.5; 

3. 738.0, 307.7 

4. (a) a - 312 (c) c - 76.0 

6 - 416 a - 68 

(b) b - 469.3 (d) 6 - 96.2 

c = 997.3 a = 231.0 

6. 68ft. 9. 37.17ft. 

6. 245.7 ft., 172.2 ft. 10. 58.4 ft. 

7. 274.7 ft. 11. 5590 yd. 

8. 66 ft. 12. 105.0 ft. 



(d) b - 19.32, a - 5.18; 

(e) a - 42.3, 6 = 90.6; 
(/) a - 21.84, c = 63.84 

(e) b = 61.2 
c = 183.5 

(/) a = 803 
c = 852 

13. 3.915 cm. 

14. 24.28yd. 



8. Pages 22, 23 

1. x - 13.5, y - 19.7, 2 - 22.5 2. 2- = 19.2, y = 14.4, z - 10 

3. s a 6, J = 5.54, w = 2.31, j = 8, y = 3.08, z = 7.38 

4. a: - 150, w = 250, 37 = 117.6, z = 220.6 

6. y - 74.27 6. /*/; = 72.14 

7. v = 2.4, w = 3.2, g = 5.52, R = 2.330, s = 2.517, t = 3.915 

9. Pages 23 to 27 



. -,, f ., f . 
V 29 V29 

2- tf , T 8 T, TV 

4. (a) cos A = ^, tan A = -y, etc.; (6) sin A = -f^ t cos A = yy, etc.; 
(c) sin A = -j^g-, tan A = ^y, etc. 

6. (a) Hi (W -Mi 7. i(3 + \/2l) 

6. 1 8. 39, 36 

9. 6 - 65, c = 57, a = 68, altitude to 6 - 52.62, altitude to a - 50.34 

11. a 12, &_= 6\/3, c ^SV^ 

12. a = 3\/34, 6 - 4 A/34, c = 5V34; |, |, |, etc, 

13. AD - 28, AO - 21, OB = 20, OC = 15, DC = 4\/130' 0; = 
sin /3 = ft, cos ft = |i, tan = |?-, etc.; sin T = f , cos 7 = 



tan y = f, etc., sin 6 = :-=> cos 5 

V130 

14. AO - 57.12 ft. 

16. CD - 12, AD = 35, AB - 30, A# = 1 
5 



9 



tan 5 = J-, etc. 
15, C5 - 13, CE - 4V34; 



> cos 



V34 
16. AD - 25, DB - 15, Atf - ^, 

sin AED - . > cos AED - 

V481 



> tan 



1 ft 



tan 



ANSWERS 



17. DA - 1, DC 



sin 15 
etc. 
18. sin 22 




2\/2 - \/3 



\/2 - -v/2 



cos 22 



2\/2 - 



_> tan ' 



2 - \/2 



19. 
21. 



2.757 cm. 
5272 ft. 



22. 184.6ft. 

23. 10.78 miles 



12. Pages 31, 

(c) cot 30'; 

(d) esc 4040'; 



(a) cos 15; 

(6) sin 3; 

20; 10; 5; 920' 

(a) cos 0; (c) esc d; (e) sec 0; 

(6) sin 0; (d) \ ; (/) cos 0; 

ll51'25f"; 628'; 435'20"; 1442' 



(e) tan 4410 / ; 
(/) sec 1939'44" 



to) 



(t) tan 



i; 



13. Pages 33, 34 

(a) cos 2 0; (c) tan 2 0; (e) -oot 2 0; (flf) 1; (A) -sin 2 tan 2 
(a) 1; (c) cot 2 v; (/) 1 



3. (a) 2 sin 3 - 2 sin 5 0; (6) 2 sin 2 0-1; (/) 



(1 - sin* 



1. 
2. 



sec 
tan 



14. Pages 35 to 37 

3. 1 

4. 1 



5. -1 

6. -1 



15. Pages 39, 40 

cos A = \/l sin 2 A, tan A = sin 



, etc. 



sin A \/l cos 2 A, tan A = \/l cos 2 A /cos A, etc. 
(a) sin A = 1/X/ 1 + c ot 2 ^ cos A =* cot A/y/l + cot 2 A, 

tan A = -- 7 etc. 
ot A 



(6) sin A = \/ sec2 ^ I/sec A, cos A = I/sec A, 
tan A = \/sec 2 A 1, etc. 

16. Pages 42 to 44 

(1 + tan 2 A) 2 - sec 2 A + tan 2 A sec 2 A 

DE = a cos A sec B, CE a sin A -f a cos A tan 

a cos 4 6. 71.88, 92.21 



a sin 4 
43.2, 75.23 



i. i s * n * 

7. tan - 

1 + cos 

8. AB = a sin 2 0, a cos* 



372 PLANE AND SPHERICAL TRIGONOMETRY 

9. FD = sin v> sin 0, CD = cos <f> sin 

10. FD = sec 6 tan ^ sin = tan tan </> 

11. sin 20 = 2 sin cos 

17. Pages 44 to 47 

1. (a) cos 25, (b) cot 41, (c) esc 8 

2. (a) cos 2 (c) 2 (e) sec 2 (0) 2 
(b) 1 (d) sec 2 (/) sin 2 

1 sin 2 A 

4. (a) . - - (c) sin A 

sin A 

(b) I/sin A (d) 1 - 2 sin 2 A 
6. (a) cos A (b) cos 2 A 

6. (a) tan (6) t:in 2 + tan 4 

7. (a) 1/siii cos (6) (I - cos 0)/sin (c) (1 -f sin 0)/cos 
9. (a) a sin (d) a sin 4 (0) 6 sin sec 

(6) b sin (c) a sin 6 (/i) 2a sin 3 sec 

(c) b tan (/) 6 esc (0 2a cos 
38. 12.68 39. 69.14, 107.5 

41. x - 14.0042, y = 21.786 

42. AC *= a sin cot <, AR = a sin cot cot a 

18. Page 49 

2. 7 4. A 

3. right angles clockwise 6. 24 right angles 
6. (a) 1; (b) 2^; (c) 8; (d) 8000; (e) ^j (/) 



19. Pages 60, 61 
3. (a) A, if, A, etc.; (6) - ,?_-r-, -_?=,, ?, etc. 



4. (a) On a line parallel to y-axis and 3 units to left of it 

6. 0; 6. (a) I; (b) II; (c) IV; (d) III 

7. (a) pos. I, II; neg. Ill, IV 

* 20. Pages 63 to 66 

1. (a) -t, -t, , etc.; (6) -f, |, -f , etc. 

3. HhA -\/3, iV3, -2 

6. (a) A, if, A, etc.; (d) -A, T, "A, etc. 

6. (a) sin = f , cos = f , tan = f , etc. 

(c) sin = -jf, cos = A, tan = -^p, etc. 
(c) sin = ^V* c s ^ ~ 1?) tan ^ "^ir* etc. 

3 2 

(0) sin = -- --=t cos = T^TI tan = 4, etc. 

-\/13 \/13 

(t) sin = 0, cos = 1, tan 0=0 

7. (a) I, II; (d) II, IV 

8. (a) II; (d) 111 



ANSWERS 373 

9. (a) sin =* f, tan f, etc. (c) cos = y^, tan = &, etc. 

(e) sin = i 8 ?-, cos = TT tan = -j^, etc. 

V3 1 

(g) cos -- > tan = -- etc. 

2 V3 

(t) sin - - &, cos = i-f , etc. 
(k) sin = yf, tan = J ^, etc. 
10. - ^ ". 3 12. - f ft 

22. Pages 67, 58 

1. -t, -4\/3, iV3, etc. 

3. (a) 30, 150; (c) 30, 210; (e) 45, 315; 
(6) 210, 330; (d) 150, 330; (/) 135, 225 

4. (a) 90; (c) 0, 180; (e) 0, 180; (0) 90, 270; (t) 0, 180 
(6) 180; (d) 90, 270; (/) 270; (h) 90, 270; 

6. (6) -0.96G, 0.259, -3.732, -0.268,_3.864, -1.035 
6. 2. 7. (a) ^(-v/3 + 2); (6) V2 + i; (c) f ; (d) -f . 8. 1 

16. (a) 3 (6) 4 (c) -2 (d) 4 

24. Pages 60, 61 

1. sin 40, - cos 40, - tan 40, etc. 

2. - sin 35, cos 35, - tan 35, etc. 

3. (a) - sin 63, - cos 63, - tan 63, etc. 
(d) sin 10, - cos 10, - tan 10, etc. 
(h) sin 70, - cos 70, - tan 70, etc. 

25. Pages 63, 64 

1. (a) - sin 85, - cos 85, tan 85, etc. 
(b) - sin 85, cos 85, - tan 85, etc. 

2. cos 5, - tan 22, - esc 23, etc. 

3. (a) sin (c) tan (e) esc (g) cot 

(b) cos 20 (d) - sec (/) - sin 20 (h) cos 

4. (a) sin 160 = - sin 200 - - sin 340 - cos 70, etc. 

(c) - tan 105 = cot 15 tari 255, etc. 
(/) cot 67 = - cot 113, = - cot 293, etc. 
(i) cot 10 = tan 80 = - tan 100, etc. 

6. (a) - sin 2 25 - cos 2 86 

26. Pages 64 to 66 
2 



T=: 
V13 



, etc. 2. cos * J-, tan , etc. 



3. (a) 210, 330 (c) 135, 315 (e) 210, 330 
(b) 60, 240 (d) 45, 315 (/) 120, 240 

4. (a) sin 75 (c) sec 20 (e) - esc 70 
(b) - cos 10 (d) cot 62 (/) tan 4 

5. (a) sin 10 (c) - tan 15 (e) - esc 10 
(6) - cos 15 (d) - tan 30 (/) - sec 5 



374 



PLANE AND SPHERICAL TRIGONOMETRY 



6. (a) 



-f\/3 



W) 



7. i(l - \/2) 
14. - 1 



\/3 -2 



(0 -V2 

(/) V3 
9. sin 80 cos 80 



16. - i(3 -f 2 V2) 
9. Page 68 



1. (a) 6.72, (b) 985, (c) 69,300, (d) 4940 

2. 49ft. 



1. 0.678 

2. 0.582 
9. 5828' 



1. b 
c 

B 

2. a 

b 



28.40 
42.78 
4135' 
40.23 
22.52 
6046' 



1. 85' 

3. 0.7178 miles 

4. 114.3 
6. 5033' 
6. 11.48 



1. A - 3652' 
B - 538' 

6 - 80 

2. B = 5120' 

c - 80.9 
6 - 63.2 

3. A - 2110' 

- 1884 
c - 2020 



1. A - 3120' 
B - 5840' 
c - 23.7 



3. 0.407 

4. 2.663 
10. 6237' 



29. Page 70 

6. 2.153 
6. 3.563 



7. 4213' 

8. 2446' 



30. Page 72 



3. Impossible 



4. A - 5027' 
B - 3933' 
c =* 3.943 



6. a = 106.2 


7. c - 45.61 


c - 125.6 


A = 640' 


A = 5745' 


B == 260' 


6. a = 22.20 


8. a - 12.76 


6 - 42.10 


6 - 34.73 


B = 2748' 


# = 2010' 



31. Pages 74 to 76 

2. 6.301 miles, 8.044 miles 

7. 6821 11. 99.0ft. 

8. 3214 12. 20.90ft. 

9. 127.2, 141.2 13. 0.1299 miles 
10. 23.34, 166.1 







3! 


2. Page 78 








4. 


B 


= 


26 


7. A 


= 


274' 




a 


= 


410 


a 


= 


24.37 




c 


= 


457 


c 


= 


53.56 


6. 


A 


= 


8348' 


8. A 


= 


4318' 




a 


= 


36.98 


B 


= 


4642' 




b 





4.02 


b 


== 


0.662 


6. 


B 


= 


4630' 


9. B 


= 


1753' 




a 


= 


7.71 


b 


= 


26.91 




b 


= 


8.12 


c 


= 


87.6 



33. Page 79 

2. A 412' 
B 4858' 
c =- 153.8 



3. A = 65 
B = 25 
c - 55.2 



ANSWERS 



.375 



4. A - 339' 


6. A = 6723' 8. A - SOW 


B - 5651' 


5 - 2237' B - 5923' 


c - 499 


c - 13 c = 82.5 


6. A = 3930' 


7. A = 45 9. A = 342' 


B = 5030' 


B - 45 5 - 8618' 


c - 44 


c - 18.67 c = 4.8 




35. Page 81 


1. 9.80599 - 10 


6. 9.95656 - 10 


2. 9.93542 - 10 


7. 9.56544 - 10 


3. 9.17665 - 10 


8. 0.55211 


4. 9.73470 - 10 


9. 0.82153 


6. 9.93499 - 10 


10. 9.98988 - 10 




36. Page 82 


1. 1154'31" 


6. 8031'59" 


2. 68'9" 


7. 5216'58" 


3. 4412 / 7 // 


8. 5357'31" 


4. 743'44" 


9. 62'28" 


6. 3329'52" 


10. 528'53" 




37. Pages 83 to 85 


1. a = 9.8030 


B = 3r33W B - 1342'28" 


c - 17.091 


c = 757.26 12. a = 193.55 


B - 55 


* 7. B = 13*23' W b = 1660.9 


2. o = 5.9407 


6 = 22.757 A = 638 ; 49" 


b - 2.0205 


A = 7636'22" 13. 30.559 ft. 


A = 7113' 


8. 6 = 18.168 14. 65.714 miles 


3. b = 810.80 


c - 39.810 16. 2964.2 ft. 


A - 4731'32" 


A = 6250'46 // 16. 019'45" 


5 - 4228'28" 


9. a = 17.350 17. 9.8768 ft. 


4. A == 7409'05" 


b = 17.854 18. 3515'51" 


B = 1550'55" 


B = 4549 / 22 // 19. 19.031 in. 


c = 9.0220 


10. A = 2938 / 28 // 20. 10,524 ft. 


6. a = 388.25 


c = 6.6550 21. 3532'16" 


6 - 548.90 


B = 6021'32" 22. 957.75 ft. 


B = 5443'35" 


11. b = 17.595 23. 99.990 ft. 


6. A - 5826'54" 


c - 74.247 24. 2957.2 miles 


25. 18'46", 8100 ft. 


26. r = 7.5492, R = 8.1710 


27. B - 4047'2" 






38. Pages 86 to 89 


1. 48.798 ft. 


4. MN a cot </> cos 2 </> 


2. 14.392 ft. 


6. AOB - 11.964 


$. x = m sin (0 <?) 


esc (a 6) cos a 


8. 4470.1 ft. 


9. 89.3 ft. 10. 272.40 ft. 



11. 864 ft., 708 ft., 246 ft. 



12. 69.768 ft. 



376- 



PLANE AND SPHERICAL TRIGONOMETRY 



13. 275.94ft. 



1. A - 3412'20" 

b - 153.00 
B . 5547'60" 

2. a - 434.16 
6 - 449.58 

B - 46 



14. (a) 20.558 miles 
(6) 39.847 miles 



39. Pages 89 to 93 

3. a - 58.239 
c = 75.330 

A = 5038' 

4. 6 = 96,915 
c = 10,904 

A = 2716'26" 



5. a = 2.2883 
6 = 5.4275 

A - 2251'40" 

6. A - 2647'26" 

c - 8.8762 
# - 6312'34" 



7. 5374.0 yd., 8302.2 yd. 

8. 4880 cu. yd. 

11. 0.71407 miles 12. 24,099 

14. h - 142.5 ft., d - 128 ft. 



9. radius 
10. 13910'4", 80.598 miles 

13. 34.151 ft. 
15. (a) 3.415 miles; (6) 6.830 miles 



16. 2822'52" 
19. 284 ft., 291 ft. 



17. 10,910ft. 



,18. 345.81 ft., 116.75 ft. 



20. 7.87 mi. 

41. Pages 95, 96 

1. (a) to (6) to W to W) T; (e) to (/) to to) to 

2. (a) 60; (6) 135; (c) 2.5; (d) 210; (e) 1200; (/) 176.40 

3. (a) 0.01745; (6) 0.0002909; (c) 0.000004848; (d) 0.1778; (e) 3.152; 
(/) 5.244 * 

4. (a) 544'; (b) 14314'; (c) 9140'; (d) 34346 / 

5. (a) \/3 (d) \/3 to) i\/3 
(6) i\/3 (e) 1 W - 2 
(c) i\/2 (/) - 1 (0 



? 

(6) i^4 

(0 fr, J 

7. (a) x - 0, y - 

(6) x - 0.36234, y - 1 

(c) x - 0.15642, y = 0.58578 

(d) .r - 3.29816, y - 3.41422 

(e) x - 4.23598, y = 3.73206 

(f) x - 8.33030, y = 3.73206 

8. (a) * - 5, y - 

(6) x = 7.03450, y - 1.71215 

9. 91 21 ; 



(d) 



to) 
(W 

0') 
(*) 



1.14160, ?/ - 2 

6.28318, y - 4 

11.42476, v - 2 
: 12.56636, y - 
i 43.98226, y - 4 



(c) x - -13.4930, y - 13.3610 



42. Pages 97 to 99 

1. (a) 226.20ft.; (c) 217.92ft.; (e) 0.13264ft.; 
(b) 358.14ft.; (d) 4.2935ft.; (/) 4o ft. 

2. (a) 36; (6) 112'; (c) 7'12"; (d) 126'24"; (e) 33650'24" 



ANSWERS 377 

4. 7.5 ft. 6. 944' 6. 75 yd. , 7. ^ 

8. 247.16 r.p.m., 25.882 radians per second 

9. 0.00098175, 1018.1 18. 17.045 miles per hour 

11. 72 yd. 19. 7.3304 ft. per second 

12. 0.015708 20. 846.40 ft. 

13. 69.088 miles, 932.71 miles 21. 222.67 ft., 4583.8 ft. 

14. 2160 miles 22. 589.33 ft. 

16. 2.2270 ft. 23. 20.944 ft., 200 ft. 

16. 62.857 radians per second 24. 294,51 ft. 

17. 1760 radians per minute 25. 2.9630 mils 

45. Pages 104 to 106 

3. sin (A + B) = sin C, cos (A + B) = - cos C, tan (A + B) - - tan C 

4. 1 + cos 0, 1 -f sin 0, hav 0, hav 9, vers 0, covers 

6. (a) - cos 10 (c) - cot 20 (e) '- tan 80 

(b) - tan 70 (d) cos 20 (/) - sin 60 

6. (a) cos (d) cos (g) sec 

(b) tan (e) tan (h) sin 

(c) tan (/) sec 

7. (a) 0.984, -0.177, -5.539, -0.180; 

(b) -0.582,0.813, -0.716, -1.397; 

(c) 0.295, 0.955, 0.309, 3.239 

8. (a) 3 (c) esc 2 (e) - cot 
(b) - 1 (d) cos 2 

9. (a) - l(\/3 + D (5) (c) 
11. i(4\/3 - 27) 12. i(2 - 3\/3) 

13. cos 2 x sin 2 x tan x 

14. (a) - 3\/3; (6) i; W i; (d) -1; () -\/3; CO ~i\/3 

60. Pages 115, 116 

1. (a) I* (e) &r (t) 3* (m) 



(c) 2r 


wj 


(W fr 


(d) frr 


(W 1 


(0 * 


2. (a) 1 


Wi 


(e) 334 


(b) 4 


(d) 8.6 


00 A 



(W 8 
10. ~> 110 

51. Pages 116 to 119 

1- ' frr, !, fir, fr, -fr,-A -0.42324 

4. 60, 180, 120, 315, 11436', 286 29', -17153' 



378 PLANE AND SPHERICAL TRIGONOMETRY 

6. (a) - tan 30 (c) - cot 36 

(6) cos 2543' (d) - esc 2543' 

6. (a) 2.4 (6) 13730' 

7. 3.3510 8. 0.42 radian 9. 18.40 miles per second 

10. 30.159 radians per second, 753.98 ft. per minute 

11. (a) 12. (a) cos 2 x - sin 2 x 
(W 2 (&) 1 

(c) 1 (c) cot 2 A 

(d) (d) 1 

(e) -3.9793 (e) - cos 2 
(/) - A/3 (/) 

to) 8 to) 1 

18. 

(c 2 - 1)^ 

19. sin (-6) = i4 cos (-*) = At tan ( 0) = "> eto - 

20. sin 




21. ^f 28. 523.6 31. 182.42 ft. 

22. -f 29. 92,800,000 miles 32. 304.10 ft. 
27. (n - 2)7r 30. 830.79 ft. 



53. Pages 122 to 124 



i. f(i + V76), i(4\/2 - 

3. iV^CVs + 1), |\/2(\/3 - D, etc. 

4. ir\/2(\/3 + D, etc. 6. 

6. (6) 0.0178 7. ff 8. i f 

9. sin 2 A = 2 sin A cos A, cos 2 A = cos 2 A sin 2 A 

11. (a) cos i/ } sin y (g) sin y, cos y 

(6) sin y, cos ?/ (h) cos x, sin x 

(c) sin !/, cos y (i) sin cc, cos a; 

(d) cos ?/, sin y (j) cos z, sin x 

(e) cos y, sin y (fc) sin y, cos T/ 
(/) sin t/, cos y 

(I) -: (cos y - sin y\ =. (cos y + sin y) 
V2 V2 

(m) -^ (cos y + sin y), 7= (cos y sin y) 
V 2 V 2 



(n) (cos ?/ + \/3 sin ?/), ^(\/3 cos y sin y) 
(o) i(A/3 cos y - sin y), |(cos y + -\/3 sin y) 

15. - p (\/3 + X/ 2 ) 24. 3 sin e - 4 em 8 

2V3 
25. 4 cos 3 - 3 cos 6 



ANSWERS 379 

55. Pages 126 to 128 

3. -(2 + V5) 

6. sin (a + ft) - - ff ; cos ( + 0) - ff; tan (a + 0) = -J, etc. 

6. sin (a - 0) - -jHrl; cos (a - 0) = -$fj; tan ( - 0) - +fgf, etc. 

7. -i 8. 3 

14. (a) sin 5.r; (6) cos x' f (c) sin x\ (d) 0; (e) cos 2x; (/) sin 2z 

15. (a) tan 5z; (6) tan 2x 

20. (a) 4 sin (0 + 30); (6) -\/2o sin (0 + 45); (c) sin (0 + 45); 
(d) 2\/3 sin (0 - 30); (e) 5 sin (0 + 538 ; ); (/) 2 cos (0 + 45) 

66. Pages 130 to 132 

-ft, A, -> A A/IO, iV\/io, 3 



4 tan x 4 tan 3 



2. VT- \/2, i^/2 4- -\/2 



6. (4 sin x - 8 sin 3 z)\A - sin 2 x. A , . A A 

1-6 tan 2 x + tan 4 x 

& T( V 5 ~ 1) 9 T2&I A TT& ~TT 



57. Pages 134 to 136 

1. (a) 2 sin 30 cos 5; (c) 2 cos 45 cos 20; (e) 2 cos 3* cos x; 
(0r) 2 sin 2x cos 

2. (a) (sin 10x sin 4a); (6) -y(os lOj; + cos 4x); 

(c) ^(cos 2o, + cos 4j cos 6.t 1); 

(d) ^(sin 15j -f sm 9x + sin 5x sin x) 

26. 2 sin [45 + $(x - y)} cos [-45 +'(* + l/)] 

27. 2 cos [45 + iU - y)] sin [-45 + ^U + y)J 
29. 4 sin 4a cos 2a cos a 

58. Pages 136 to 139 

2. (a) ff 6. (a) f 
(0 (6) * 

39. Varies from to 1 

45. 1 - 18 sin 2 a + 48 sin 4 a - 32 sin 6 a 

69. Pages 142 to 144 

1. x . y 4\/3, * - 18, y 31.172 

j a; - 35 sin 60 esc 70; Fig. 6: a = ?/ = 35 sin 70 esc 40 
2 - ** 5 J ?/ 35 sin 50 esc 70; Fig. 7: j; = 40 sin 11120' esc 30 
Fig. 8: x - 60 sin 7425' esc 40, y = 60 sin 2535' esc 40 

3. x = esc 30 sin 80, y = esc 30 sin 50, z = esc 30 sin 50 sin 80 esc 60, 
p = esc 30 sin 50 sin 40 esc 60 

4. sin B = 0.68627, x - 624 sin (118 - B) esc 62 

5. [312 sin (118 - #)(csc 62)] 485 sin 62 

6. x = a sin 65 esc 40, y = a sin 75 esc 40, x = a esc sin (b + *>), 
y = a esc sin ^ 



380 



PLANE AND SPHERICAL TRIGONOMETRY 



7. x sin 50 esc 60, z = sin 50 esc 30, w -- 
y - sin 2 50 esc 60 cse 70 

61. Pages 148, 149 

6 sin 60 8 sin 60 



sin 50 esc 70, 



V52 ' V52 

3. Fig. 23: x 

4. i tan 45 



2. tan (A - B) 



' /=- . A 3 sin 30 . 

15\/3, sin A = sin 

JL 



+\/3 
5 sin 30 



62. Pages 149 to 151 

1. V 1873 - 924\/2 2. / T tan 67^ 3. 462 sin 45 

c 2 sin A sin 7? n 

5 ' AfW "2rin<A+lT) 6 - 

7. (a) 8 sin 60 sin 40 cse 50 esc 35 (6) 10.136 

8. h = m sin w esc (w + z) sin y esc (a; + y) 

66. Page 165 



1. 6 = 4.4217, 


c = 1.7302, C - 


2224' 


2. 6 - 4382.9, 


c - 6136.0, A = 


8147'12' / 


3. a = 895.14, 


b = 728.40, C - 


6734 / 31 // 


4. a = 177.64, 


6 = 213.78, B = 


6219 / 53" 


6. a - 241.18, 


6 = 165.68, C - 


6812'15' / 


6. b = 695.32, 


c - 345.64, C = 


2114'20" 


7. 345.43 






8. 73.548 ft. 


12. 22322ft. 




10. (a) 3.113 


13. 590.43 ft. 




11. 26,624 ft., 26,689 ft. 


14. 192.41 ft 






66. Pages 160, 161 




1. Bi = 2457'54" 


B 2 = 1552'6" 




Ci = 13347'4l" 


C 2 = 343 / 29" 




ci - 615.67 


c 2 = 55.410 




2. A! = 13418'3" 


A 2 = 38 / 29" 




Ci = 2425'13" 


Ci = 15534'47" 




ai = 623.19 


o 2 = 47.718 




3. B l - 519'6" 


5 2 = 12850'54" 




Ci - 8737'54" 


C 2 = 956'6 // 




ci - 116.82 


c 2 = 20.172 




4. ai - 167.64 


o 2 - 35.124 




Ai - 8139 / 07" 


A 2 = 1157'49" 




Ci = 5509'21" 


C* = 12450'39" 




6. B 36 26'46" 


6. a - 31.672 7. J 


B = 2612'38" 


C * 761'14" 


C - 90 C - 117 23 ; 22" 


c 308.73 


A = 2347 / 50 // 


c - 72.022 



ANSWERS 



381 



8. ci - 60.303 


c 2 - 24.561 


B l - 5620'08" 


B 2 - 12339 / 52 // 


Ci - 9121'22" 


C 2 - 2401 / 38 // 


9. ci - 3.7834 


c 2 - 2.1960 


B l - 7912'00" 


J5 2 - 10048'00" 


Ci - 4630W 


C 2 - 2454 / 00" 


10. J5j - 4523'28" 


B 2 - 13436'32' / 


Ai - 9900'12" 


A 2 = 947 / 08' / 


01 - 300.29 


a 2 - 51.670 


11. Impossible 


13. 17,091 14. 4747 / 36" 




67. Pages 163, 164 


1. A 7712'53" 


4. A = 4028'17" 7. A - 5210'33' / 


= 4330'7" 


B - 9951 / 43 // /? = 1717'27" 


c - 14.987 


c - 27.458 c = 7.3962 


2. A - 8623'9" 


6. B = 5157 / 20" 8. A - 4649'58" 


= 301'21" 


C - 7722 / 16 // ' B = 2229'32" 


c - 671.27 


a = 83.732 c - 45.198 


3. B - 6737 / 44" 


6. A - 9251 / 28 // 


C = 519'16" 


B = 2230 / 32 // 


a = 220.10 


c - 0.53660 


10. 5119.5ft. 


14. (a) 87.690 


11. 147.96ft. 


16. Not horizontal; 5281.7 ft. 


12. 4064.1, 16553'45" 


17. 443.19 ft. 




69. Pages 168, 169 


1. A - 10646'40" 


6. A - 2746'44" 9. A - 80.4 


B - 4653'14" 


B - 3346 / 52" B - 56.6 


C - 2620'6" 


C - 11826'20" C = 43.0 


2. A - 2720'32" 


6. A = 5153'12" 10. A - 46.6 


B - 1437 / 48 // 


- 5931 / 48 // B = 58 


C - 931'40" 


C - 6835W ; C = 75.5 


3. A 820'1" 


7. A - 28 6'52 ;/ 11. A = 106 


5 = 3340'5" 


B - 115 b 2 ; 4 ;/ B = 39.8 


C - 13759'54" 


C - 3651'8" C = 34.1 


4. A 4442'16" 


8. A - 4537 / 18 // 13. 72.6 


B - 4937 / 26 // 


B - 75 19'32" 14. 495.53 ft. 


C - 8540'24 ;/ 


C = 593'10'' 




71. Pages 170 to 176 


1. A - 4049 / 36 // 


2. A - 4147'45 // 3. C = eQMS^S'' 


B = 2331'24 // 


B - 5420'09 // 6 = 462.76 


c - 58.416 


C - 8352'05" c - 499.00 


4. A 5210 / 33" 


5. A = 4656 / 24" 


5 - iTnrw 


B = 5711 / 08" 


e <= 0.073964 


C - 7552 / 32" 



382 PLANE AND SPHERICAL TRIGONOMETRY 

6. B! - 5656'56" B 2 = 1233'4" 
Ci - 9045'4" C 2 - 2438'56" 

ci - 58.456 c, - 24.382 

7. AC - 1474.0 ft., C - 1252.7 ft. 

8. 6328.7 ft. 9. 848'12" 10. 722.18 
12. 52.431 16. 3.1959 miles per hour 
16. 373 ft. 17. 731.13 ft., 5038' 

18. 6463.0 ft. 19. 88.016 ft. 21. 8.0126 nautical miles 

22. 444'25 32. 2109.8yd. 

23. 231.94 ft., 328.93 ft. 36. 509.77 yd. 
30. 2554.7 ft. 37. 107.24 
42. PB = 403.68, PA = 140.89, PC = 734.98 

46. 79.4 yd., 149'1" 

R 

46. [0 sin" 1 (sin 8 cos <?)], tan" 1 (tan 6 sin v>), where sin" 1 (tan" 1 ) means 


angle whose sine (tangent) is 

* 72. Page 178 

1. 30, 150 6. 135, 315 9. 60, 300 

2. 60, 120 6. 120, 240 10. 210, 330 

3. 225, 315 7. 135, 225 11. 60, 120 

4. 60, 240 8. 45, 315 12. 2536', 15424' 

74. Pages 180, 181 



A. w - -r ^/"r, -r ^ 
o o 


r/*7T 


v-y 3 i 


r ^^TT, ~r A^/I 
o 


T 27T 








(6) - + 2r, - + 2 


\n-a- 


(e) 2n7r, 


TT -f 2n7r; (or nir) 


3 3 








IT 3ir 




STT 




(c) - + 2nir, T + 2 


\mr 


(/) y 4 


- 2mr 


4 4 








(g) 1928' + n360, 


16032' + 


n360 




(h) 2536' + n360, 


15424' + 


n360 




(0 20437' + n360 


, 33523' - 


Hn360 




If iTT 




STT 


7ir 


4 ' 4 


2/iir 


(n) -- 
4 


f 2n7r, -f 2n7r 


37r 5?r 




_ 






2mr 


(o) 1 + 


nir 


4 n7F> 4 








5lT 77T 




7T 


STT 




2n* 


(P) 4 + 


nw, 4 


(m) J + 2 T , ^ -\ 


- 2mr 


(9) riTr 




6 6 








(r) 6638' + n360, 


24638 ; + 


w360 




II 


3ir 




K_ 


2. (a) + 2nir 


(0 J + 


2n7r 


(c) -- + 2nir 


6 


4 




o 


7ir 


STT 




STT 


(6) h 2nir 


W T + 


2f*r 


3 



ANSWERS 



383 



3. 


(a) 


21 6' - 


f n360, 


15854 7 - 


f- n360 






(b) 


538' - 


h n360, 


30652 ; - 


h w360 






(c) 


4159' 


+ n360 


, 221 59' 


-f n360 






(d) 


2528' 


4- n360 


, 205 28' 


4- w360 






(e) 


730' - 


f n360, 


2870 7 4- 


n360 






(/) 


5544' 


4- w360, 12416 / 


4- w360 






(0) 


538' - 


f n360, 


30652' - 


h n360 






(W 


4149' 


+ n360 


, ISS !!' 


+ n360 






w 


5120' 


4- n360 


, 23120' 


4 n360 






(J) 


4811' 


-f n360, 31l49' + w360 




(W 


4849' 


4- n36C " 


, 22849 / 


4 n360 






(0 


349 / 


4- n360, 


176ir - 


h n360 




6. 


(a) 


30 4- 


fc60 




(c) 


2722' 4- A 




(W 


*36 






(d) 


20 4- fc60 ( 


6. 


(a) 


45 -f 


fc!80 




(c) 


135 + kl\ 




(6) 


30 4- 


A;180 




(d) 


1853' 4- * 



/c!80 



75. Pages 183, 184 



i. 


(a) 


JT 


(/) 





(k) & 




(p) 




(W 


* 


to) 


^TT 


(/) *r 




() i* 




(c) 





( 


7T 


(m) ir 




(r) ir 




(d) 


IT 


W 


-K 


(n) i^r 








(e) 


i* 


0) 


-^ 


(o) i^r 






2. 


(a) 


-30 




(c) 


-60 


(e) 


-60 




(&) 


-45 




(d) 


-45 


(/) 


-30 


3. 


(a) 


135 




(c) 


120 


(c) 


150 




(W 


150 




(d) 


135 


(/) 


120 


4. 


(a) 


-fr 




,(d} 


^^ 


to) 


fir 




(b) 


-f- 




(e) 


6 71 " 


(h) 


?r 




(c) 


7T 




(/) 


fir 





^TT 


5. 


() 


-30 


(d) 


90 


to) -45 




0') -13 




(b) 


45 


(e) 


-135 


(h) 60 




(*) 180 




(c) 


150 


(/) 


-180 


(t) 60 






e. 


(a) 


-60 




(d) 


11129' 


to) 


-415 7 




(b) 


11427' 




(e) 


11516' 


(^) 


15555' 




(c) 


-5444 7 




(/) 


-171 !' 


(i) 


-8536' 


7. 


(a) 


TT 




(c) 


^7T 


(e) 


IT 




(b) 


Jir 




(d) 


-i 


(/) 


-7T 



1. 

2. 
3. 

5. 
6. 
7. 



-f 



77. Pages 187 to 189 

8. -f 14. 

9. 2/-\/5 ' * 15. 

10. i\/5 16. (o) -| 

11. 1 (b) 2/\/3 

V^SO.IG (c) 1 

5.4 W -0.993 

13. 



384 



PLANE AND SPHERICAL TRIGONOMETRY 



'14-64- (2 2 - 



78. Pages 190 to 192 

1. (a) 30, 150, 210, 330 (d) 60, 120, 240, 300 
(6) 45, 135, 225, 315 

(c) 60, 120, 240, 300 

2. (a) 120, 240 

(6) 30, 150, 210, 330 

(c) 60, 120 

(d) 60, 300 

3f~\ 1 3 4 
() -jr^, T 71 "* ~a 



T 71 " 



(e) 22, 112, 202^, 292^ 

(/) 10, 50, 130, 170, 250, 290 

(e) 30, 150, 210, 330 

(/) 45, 225 

(0) 135, 315 



, fr, 



4. (a) n360, 120 4- n360, 240 4- w360 

( 30 4- n360, 150 + n360 (c) 270 4- w360 

(d) 45 4- n!80, 105 4- nl80, 165 + n!80 

to 5619' 4- n!80, 135 4- n!80 

(/) 3341' 4- n!80, 45 4- "180 

(g) 3759' 4- "45 

(fc) 90 4- n!80, 60 4- n!80, 120 4- n!80 

(i) 5119 ; 4- n360, 30841 ; 4- n360, 180 4- n360 

(j) 30 4- w360, 150 + w360, 90 4- n360 

(A;) 45 4- n90 (/) 45 4- n!80, 7134' 

(m) 120 + n360, 240 4- w360 

(n) 944' + n360, 1512r 4- n360 

(0) n360, 90 + n360 (p) 60 4- n360 
(q) 105 4- n!80, 165 4- n!80 

(r) 90 4- nl80, 120 + n360, 240 4- n360 
(s) 30 4- n!80, 150 4- n!80 
6. (a) n!80, 60 4- n360, 

(6) 90 4- n!80, 30 4- n360, 150 4- n360 

(c) n!80, 60 4- n!80, 120 4- n!80 

(d) 90 4- n!80, 210 4- "360, 330 4- w360 

(e) 45 4- w90, 15 4- n!80, 75 4- ^180 
(/) 30 4- n360, 330 4- "360, w!80 

(g) n90, 30 4- "90, 60 4- w90 

(h) n90, 5214' 4- n!80, 127 46' 4- n!80 

(1) n!80, 60 4- w!80 

6. (a) mr (c) nir 

(b) 2n7T, -|ir 4- 2n7r, -Jvr 4" 2nir (d) mr 



n!80 



1. (a) n60, 15 4- w30 
(6) n45 



79. Pages 194, 195 

(c) 5 4- "20, 22^ 4- n90 



(c) 9 4- n!8 

(/) 45 4- nl8Q, 6 4- n20 



ANSWERS 



385 



(g) -2520' + n360, 13136' + n360 
(h) 90 + n360, 19616' -h w360 
() 14237 / + n360, 26237' + n360 
(j) 88' + w3GO, 2176' + w360 
(A;) 135 + n!80, 161 34' + n!80 
(o) = n45, 12 + n72 



(*) x 






3. tan (x -h 



, = 7134' + n360; a; = X/10, = 25134' + fc360 

" 54 12 ' + w360 ' * = 56 19/ + n360 
= 12 548' -f n360, == 23619 / + "360 

= -5412' + n360, - 23619' + ^360 
^ = _i2548 ; + w360, = 5619' + n360 

tan - which determines x -f cr, and therefore x 



_ 

6 sin 
cos 

m = [a^ -f fc 2 - 2afe cos (^ 
b cos a sin < 



\ a sin ^ 6 si 
4. x - tan" 1 - ------- 

|_6 cos a cos 



0) 



5. m sin x 



m cos x 



tan" 1 . 



cos (0 <t>) 
b sin + a cos < 

cos (0 <f>) 
b cos a sin 



b sin + a cos 
m = [a 2 + b 2 2a6 cos (0 + 

6 cos a cos 

6. m sin re = . - - > 

sin (0 -f 0) 

6 sin + a sin 

m cos x = . ; x ~ 

sin (0 + 0) 

6 cos 0o cos 

x = tan l -- .- r 

b sin -f- o sin 

w = [ a 2 _|- ^2 _ 2a6 cos (0 -f 

7. x = m cos a + n sin a 



sec (0 - 



esc (0 + 0) 
?/ = w sin a n COS a 



2. (a) y 

(b) 1 

(c) i 

w) a ' 



i\/5 



80. Pages 196, 197 

to) 



1 - a 2 - i 

() 0, i\/3 
(/) No solution 




m > 0, n > 0; 
t 8 , m < 0, n < 0. 



386 PLANE AND SPHERICAL TRIGONOMETRY 

81. Pages 197 to 199 

1. (a) A. (c) ^- (e) 2a 2 - 1 (g) nir + - 



(6) -p (d) A 0) -7== (*) n - 

V2 V<* 2 + 1 4 

3. (a) 7134' + "360, 251 34' + n360 
(6) 15832' n360, 201 28' n360 

(c) n!80 

4. (a) 19928' + n360, 34032' + n360 
(6) 7032' -f n360, 28928 ; + w360 

(c) 45 + wl80, HG^^ + wl80 

(d) 210 + n360, 330 + n360, 4149' + n360, 138 !!' + n360 

(e) 90 + n!80, 210 -h n360, 330 + w360 
(/) 20428' + w360, 33532' -f n360 

(g) 7640' + n!80, 3473' -h n!80 
(h) 135 -h n!80 

(i) = 270 -f n360, 12652' -f n360 
(j) n360 
(*) 60 + n360 
(/) 30 4- n90, 3516' + n90 
6. (a) n90 (c) n!80, 30 + n90, 60 + n90 



7. (a) n360, 10616' + n360 (6) 7720 ; + "360, 180 + n360 

8. (a) 240 + w360, 300 + w360 
(6) 210 + n360, 330 + n360 

(c) 30 - n!80 

(d) 4921' + w360, 31029' + n360 

(e) 60 + n720, 300 -h w720 

9. (a) n90, 120 -f ^360, 240 + n360 
(6) n60, 3516' -f n!80 

(c) 30, 90, 150, 210, 270, 330 (add n360 to each) 



10 (d) (x - a] 


i' (y-c] 


1 (e) ( b ) 


( X \ 1 


11. (a) i 

(6) V/3 


(c) + 
(d) V 


w - 1 

( n \ 1 ^ 


^ (/) i, 


14 
W 13 






82. Page 200 




1. (a) 6i 
(6) 3\/3* 


(c) 7 

( V& 


(e) 4xi 

2 

-i (/) ~i 


(0) 5x 2 y-\/5i 
(/i) i\Aac - 



ANSWERS 387 



2. (a) 4i; (6) 3rri; (c) \/13i; (d) a*x\/7i 

3. (a) t; (6) 1; (c) - 1; (d) - 1; (e) - i; (/) 1; fo) -1; (*) 1 

84. Pages 201, 202 

1. (a) x - 2, y - -3; TC) * - f , y - 4; (e) * - -1, y - 
(6) * - I, y - ; (d) x = 3, y - I; 

2. (a) 7 - ft; (6) * + yt; (c) -3t; (d) 14 

3. (a) 5 - i (c) 6 - 3t (e) 6 fo) 2 - ft 
(6) -4 + 8t (d) 3 + 4t (/) 3 -f 7* (A) 8t 

6. (a) 28 + 24i (c) 2 + 16i (e) 5 + ft 
(6) 20 - 48t (d) 65 (/) 32 - 26t 

7. (a) f| - Jit (d) A + tM ((7) II - i 

(6) t - it 4 - fit w -iH- + in* 

(c) ii -h A* V) -A + A* W 0.02 - 0.64t 

86. Pages 204, 205 

1. -3\/3 - 3t, 6(cos 210 + i sin 210) 

2. \/2(cos 45 + i sin 45) 

3. (a) \/3 + 1 (d) -4 ((7) 3 - 3\/3 
(6) | -h l\/3 ( e ) ~ V-V3 - -i (^) 3 - 3\/3t 
(c) -V^ + \/2t (/) -7t 

4. (a) \/2jcis315 (t) 1.5 cis 270 
(6) \/13 cis 23619' (/) 1 cis 270 

(c) Vl3 cis 123 41' (k) 8.60 cis 32428' 

(d) 4 cis (0 6.28 cis 30039' 

(e) 4 cis 90 (m) 7.41 cis 14528' 
(/) 5 cis (n) 7.38 cis 24326' 
(g) 7 cis 90 (o) 8.35 cis 32813' 
(h) T/IJ cis 5732' 

88. Pages 207, 208 

1. (a) 9.70 + 17.5t (c) 2.16 - 2.08i 
(6) 4.86 + 27.6t (d) -0.82 + 1.13i 

2. (a) t (e) -0.518 + l-93t (h) 0.518 + 1.93t 

(b) \/2i (/) 4.39 -f 16.4t (i) 0.185 + 2.1H 

(c) 2\/2 (0) 0.228 - 0.0610t (j) -0.958 + 0.804t 

(d) -2 

3. (a) -7.42 -h 12.86t (c) 6i 

(6) -0.101 + 0.175t (d) -50.9 - 88.2t 

4. (a) 5 cis 280 (6) 5.54 - 5.28t (c) 17.1 - 38.4i 

89. Pages 210, 211 

1. (a) 16 cis 120 (c) cis 30 (e) $72\/2 cis 135 

(6) 4 cis |TT (d) cis 180 -- 1 (/) i ois 180 



388 PLANE AND SPHERICAL TRIGONOMETRY 

2. (a) 3.44 cis 34431', 3.44 cis 16431' 

(6) cis 60, cis 132, cis 204, ois 276, cis 348 

(c) cis 18, cis 90, cis 162, cis 234, cie 306 

(d) cis 60, cis 180, cis 300 

(e) 1.74 cis 7658 / 1.74 cis 16858', 1.74 ci25658', 1.74 cis 34658' 

(/) 1.341 cis 5, 1.341 cis 45, 1.341 cis 85, 1.341 cis 125, 1.341 cis 165, 
1.341 cis 205, 1.341 cis 245, 1.341 cis 285, 1.341 cis 325 

(0) cis 20, cis 60, cis 100, cis 140, cis 180, cis 220, cis 260, cis 300, 
cis 340 

3. (a) x = -1, x = 0.5 0.8661 

(6) x - -2, x - 1.62 t.lSi, x = -0.618 1.89t 

(c) x - i, x = 0.866 - 0.5i 

(d) x - 0.855 1.48t, x = 1.71, x = 1.913, x = -0.956 1.66i 

(e) x = 1, x = 0.707 0.707i, x = -0.5 0.866i 

90. Page 212 
1. -1, t, -0.41655 + 0.9091H, t 2. 3.7622, -3.6269i 

91. Pages 213, 214 
1. 1, 0, 1.5431, 1.1752 

92. Page 214, 215 

1. (a) |\/2 + Iv^i () 2.64960 + 4.24025z 
(6) -2\/3 -f 2i (/) -4.47352 + 6.63232t 

(c) | - | V& (g) 4.85412 - 3.52674i 

(d) 7i (ft) -1.52458 - 1.29446* 

2. (a) 2\/2 cis 45 (/) \^\ cis 30948' 
(6) 3\/2 cis 315 to) 2^/10 cis 341 34' 

(c) VlO cis 16134' (ft) 4 cis 21652 ; 

(d) \/13 cis 30341' (i) \/1^6 cis 161 34' 

(e) 5 cis 2338' 

3. (a) 14 cis 210 (6) 3.2966 cis 1413' (d) 10.181 cis 15926' 

4. (a) 32 cis 225 (r) 16 cis 120 

(b) (2.6) 3 cis 219 (d) 5 5 cis 27420' 

5. (a) 1.4142 cis (-15) (d) 1.4554 cis 1253' 

1.4142 cis 165 1.4554 cis 8453' 

(6) 1.4953 cis (-913') 1.4554 cis 15653' 

1.4953 cis 8047 ; 1.4554 cis 22853' 

1.4953 cis 17047 ; 1.4554 cis 30053' 

1.4953 cis 26047' (e) cis (-30) 

(c) 1.8301 cis 7846' cis 90 
1.8301 cis 19846' cis 210 
1.8301 cis 31846' 

6. (a) 2, -1 \/3i (6) i<\/3~ +ii, -i 



ANSWERS 



389 



(c) 1.3077 cis -851 ; 
1.3077 cis 51 9' 
1.3077 cis 1119' 
1.3077 cis 1719' 
1.3077 cis 2319' 
1.3077 cis 291 9' 



1. 5. 2 

2. 5 6. 1 

3. 1 7. 8 - 10 

4. 8. 9 - 10 



(d) 1.3446 cis 3430' 
1.3446 cis 8556' 
1.3446 cis 13722' 
1.3446 cis 18848' 
1.3446 cis 24014' 
1.3446 cis 291 40' 
1.3446 cis 3436' 



7. Page 223 

9. 4 

10. 2 

11. 5-10 

12. 7-10 



1. 1.60733 

2. 0.48391 

3. 4.00864 

4. 2.03411 



1. 0.04592 

2. 7903 

3. 207,320 

4. 0.50119 

11. (a) 0.45347 
(6) 0.0038615 



101. Page 226 

6. 9.33333 - 10 

6. 7.58371 - 10 

7. 8.93677 - 10 

8. 5.88152 - 10 

102. Page 227 

5. 0.0093962 

6. 997.15 

7. 7.4962 

8. 2.6448 

(c) 0.00074363 

(d) 0.68973 



13. 3 

14. 4 

15. 9-10 

16. 6-10 



9. 8.43198 - 10 
10. 9.26133 - 10 



9. 12.954 
10. 0.00035304 



1. 433.90 

2. 224.09 



3. (a) 5.0187 
(ft) 147.54 



1. 8.5398 

2. 0.010894 

3. 33,451 

4. 1019.4 

5. 200,530 

6. 0.19835 

7. 24.682 

8. 17.843 

9. 0.65684 

10. 0.0067010 

11. 437.88 



3. 3.1414 

4. 1.3205 



103. Page 229 

5. 0.51514 

6. 5.2686 



7. 0.24406 

8. 0.062086 



104. Pages 229, 230 

(c) 0.00041391 

(d) 5058.6 



106. Pages 232 to 234 

12. 3.1414 

13. 18.636 

14. 0.72132 
16. 0.26868 

16. 0.39770 

17. 0.39510 

18. 1.2390 

19. 1.1605 

20. 0.53670 

21. 107.42 

22. 3630.8 



23. 1.6478 

24. 3463.4 

25. 27.278 

26. -22.582 

27. 15.353 

28. 0.00021360 

29. 18.666 

30. -22.302 

31. -1.2552 

32. -5.2060 



390 



PLANE AND SPHERICAL TRIGONOMETRY 



33. 0.0074500 

34. 1.56026; (-)l. 46098; 9.05621 - 10; 2.08309 

35. 46.693 38. 266.46 11). 

36. 8.6458 39. 2283.2 It). 

37. 0.028375 40. 6.2691 ft. 

44. Volume = 13,330, Surface = 2719. 

45. 1051 X 10 7 47. 834,200. 

46. 11,660. 48. 1,476,000. 



41. 151,370 gal. 

42. 1.01 sec. 

43. 142.5 tons 

49. 0.608. 



108. Pages 236, 237 



3. 

4. 
5. 



1. 2.3666 

2. -90.006 
-1.7354 
-1.9034 
1.5372 

6. 4.9168 

7. -0.15421 

8. -0.76206 

9. 6.0110 

19. 0, 1.3169 

20. 3.96 

21. 0.00003772 



10. 1.7895 

11. 339.86 

12. 2.7183 

13. 0.42767 

14. 0.41639 
16. 0.11699 

16. -0.37979 

17. x = 3.0484, y 

18. 17.677 



22. 18,360 



25. x 



2.0484 



e* - 1 



23. k = 0.126 

24. 5.5 minutes 

110. Pages 239, 240 



26. x = 25 and -4 



1. 222.91 8. 4.4787 15. 34.801 22. 0.031072 

2. 0.037367 9. 3.0675 16. 67.535 23. 4.6249 

3. 72.888 10. 0.00079018 17. 42.620 24. 3.5064 

4. 0.0093936 11. 0.37665 18. 2362.9 25. 1.5509 
6. 24.491 12. 0.28926 19. -4.2098 26. 0.036016 

6. 1.2142 13. 0.96048 20. -0.86048 

7. 12.377 14. 1.7867 21. -0.21423 

27. (a) 0.093180; (6) 168.20; (c) 0.44668 

28. 35.239 29. 4.251 

30. (a) 100,100; more accurate value 100,081; (6) 85,450; more accurate 
value, 85,442 

31. 1547 miles 32. 146,700 sq. km. 



1. 6 

2. 7 

3. 10 



1. 15 

2. 15.8 



4. 9.1 
6. 6.75 
6. 9.62 



3. 3530 

4. 42.1 



115. Page 245 

7. 49.8 

8. 340 

9. 47.0 

116. Page 246 



10. 0.0826 

11. 3220 

12. 0.836 



13. 9.86 

14. 3.08 



5. 0.001322 

6. 1737 



7. 9.98 

8. 1,340 



ANSWERS 



391 



1. 2.32 

2. 165.2 

3. 0.0767 



1. 36.7 

2. 8.35 

3. 0.0000632 

4. 3400 



117. Page 247 

4. 106.1 6. 77.5 

6. 0.000713 7. 1861 

118. Page 248 

6. 0.00357 9. 0.01311 

6. 13,970 10. 2.36 

7. 1586 11. 0.0414 

8. 0.0223 12. 2460 



8. 26.3 

9. 1.154 
10. 0.0419 



13. 249 

14. 0.275 

15. 0.1604 

16. 0.0977 



1. x - 5.22 

2. x = 2.30, y = 31.8 

3. x = 51.7, y = 3370 

4. x = 3.97, y = 9.84, z 



119. Page 260 

6. x = 1.586, y = 41.4 

7. x = 106.2, y = 30.4 

8. x = 0.1170, y - 0.927 
0.272 9. x = 186, y = 13.42, z = 50.3 



6. 



0.1013, z = 0.0769 



120. Page 251 
1. 10,570 3. 0.0337 6. 73,100 7. 0.002224 



2. 92,200 



1. 0.001156 

2. 1.512 

3. 1.015 

4. 17.2 



4. 1.765 



6. 249,000 8. 0.314 



9. 1.799 
10. 0.1555 



121. Page 253 

6. 96.1 9. 9.76 

6. 0.1111 10. 0.00288 

7. 150,800 11. 144,700 

8. 15.32 12. 0.0267 

122. Page 254 



13. 0.279 

14. 41.3 
16. 111.1 
16. 3430 



1. 2.83, 3.46, 4.12, 9.43, 2.98, 29.8, 0.943, 85.3, 0.252, 252, 316 

2. (a) 231 ft., (b) 0.279 ft., (c) 5720 ft. 

3. (a) 18.05 ft., (6) 0.992 ft., (c) 49.7 ft. 



1. 64.2 

2. 11.41 



2. 



3. 



4. A. 



3. 1092 

4. 0.428 



123. Page 255 

6. 9.65 
6. 0.0602 

124. Page 267 



7. 1.525 X 10* 

8. 1.589 



B. 



6. 



(a) 0.5 


(c) 0.0581 


(e) 0.999 


(g) 0.253 


(i) 0.204 


(b) 0.616 


W) 1 


(/) 0.0276 


(/i) 0.381 


(j) 0.783 


(a) 0.866 


(c) 0.998 


(e) 0.0393 


(g) 0.968 


(0 0.979 


(b) 0.788 


(d) 


(/) i.oo 


(h) 0.924 


(j) 0.623 


(a) 30 


(c) 222' 


(e) 51 '34" 


(g) 333' 


(i) 6656' 


(6) 61 6' 


(d) 544' 


(/) 3819' 


(h) 146'34" 


0) 6215' 


(a) 60 


(c) 6758' 


(e] 898'2ti" 


(g) 8627' 


(t) 234' 


(b) 2854' 


(d) 8416' 


(/) 5141' 


(h) 8813'26" 


(j) 2745' 


(a) 2 


(c) 17.21 


(c) 1.001 


(g) 3.95 


(t) 4.90 


(b) 1.623 


W 1 


(/) 36.2 


(h) 2.63 


0') 1-277 



392 


PLANE AND SPHERICAL 


6. 


(a) 1.155 


(c) 1.002 


(e) 25.5 




(6) 1.27 


W - 


(/) 1 


7. A. 


(o) 30 


(c) 36 


(d) 924' 




(b) 2438' 






B. 


(a) 60 


(c) 54 


(d) 8036' 




(6) 6522' 







to) 1.033 

(h) 1.082 

(e) 043' 

(c) 8917' 



(t) 1.021 
0') 1.605 



7746' 



125. Page 258 



1. 0.1423, 0.515, 1.906, 0.01949, 3.55, 19.08, 1.09 
7.03, 1.942, 0.525, 51.3, 0.282, 0.0524, 0.917 



2. (a) 1330' (d) 2822' to) 2322' (j) 2030' (m) 8638' 
(6) 388' (e) 323' (h) 228' (A;) 7457' (n) 4551' 
(c) 4237' (/) 442' (t) 51 '13" (/) 7755' (o) 5056' 
3. (a) 7630' (d) 6138' to) 6638' (j) 6930' (m) 322' 
(b) 5152' (e) 8637' (h) 8732' (A;) 153' (n) 449' 


(c) 4723' ( 


;/) 8518' (t) SW47" (I) 125' (o) 394' 




126. Page 259 




1. 30.5 


7. 5.29 13. 2.033 


19. 38.1 


2. 0.360 


8. 254 14. 0.720 


20. 0.00319 


3. 4.61 


9. 0.0679 15. 4.24 


21. 0.001091 


4. 24.2 


10. 0.267 16. 1.226 


22. 5.08 


5. 14.25 


11. 1.349 17. 0.0771 


23. 0.01375 


6. 16.79 


12. 16.47 18. 0.0961 


24. 0.0433 




127. Pages 261, 262 




1. C - 75 


4. A - 247' 7. C = 5520' 


10. Impossible 


6 = 35.46 


B - 8713' 6 = 568 


11. B - 303' 


c - 53.3 


c - 4570 c - 664 


C = 90 


2. C - 55 


5. B - 3516' 8. b - 279 


6 - 5.01 


b - 70.7 


C - 84 44' c = 284 


12. c = 123.8 


a - 56.1 


c . 138 C = 10050' 


B - 318'35" 


3. C = 12312' 


6. A - 1741' 9. A - 8741' 


C - 116 41'25" 


b - 2257 


C - 5319' C - 4112' 


13. 1253 ft. 


c = 2599 


a - 0.0751 o - 116.9 


14. 1034.8yd. 


15. B! = 6610' 


17. Ai - 7012' 19. 


Bi - 4516' 


Ci - 5826 / 


Bi = 5724' 


Ci - 998' 


ci - 18.6 


61 = 28.79 


ci - 300 


B 2 - 113 50 ; 


A 2 - 10948' 


B 2 - 13444' 


C 2 - 1046' 


B 2 - 1748' 


C 2 * 940' 


c 2 = 4.08 


62 = 10.45 


C 2 - 51.1 


16. Bi - 1643' 


18. Ai = 6847' 20. 


Ai =* 51 19' 


Ai = 14728 / 


Ci - 67 10' 


Ci - 8841' 


ai = 35.5 


ai - 6.92 


d - 21,850 


B 2 - 16317' 


A 2 - 237' 


A 2 - 12841' 


A 2 - 054' 


C 2 - 112BO' 


C 2 1119 / 


o 2 - 1.04 


a 2 - 2.91 


c 2 - 4290 




21. p - 3.13; (o) none, (6) 2, (c) 


1 



1. A - 3120' 
B = SS'W 

c - 23.7 

2. A = 412' 
B - 4858' 

r = 153.8 

3. A - 65 
B = 25 

r = 55.2 



1. 



2. 



3. 



= 119 54' 
= 3 IT/ 
= 52.6 
= 494 X 
= 797 ; 
= 104.1 
= 552' 
- 4021' 
= 285 



10. 10 and 4.68 



1. A = 10647' 
B = 4653' 
C = 2620' 

2. A = 2721 r 
B - 1438' 
C = 932' 



1. (a) 0.785 (b) 1.047 
(/) 2.36 (g) 0.393 

2. (a) 60 (c) 2.5 
(6) 135 

3. (a) 0.01745 
(6) 0.0002909 

4. (a) 544' (b) 143 



ANSWERS 




128. Page 263 




4. A = 339' 7. 


A - 45 


B - 5651' 


B - 45 


c = 499 


c - 18.67 


5. A - 3930' 8. 


A = 3037' 


B = 5030' 


B - 5923' 


c = 44 


c - 82.5 


6. A = 6723' 9. 


A - 342' 


B = 2237' 


B - 8618' 


c = 13 


c = 4.8 


129. Page 264 




4. B = 3916' 7. 


A = 1214 / 


C = 7844' 


C - 226' 


a = 3.21 


6 = 0.0828 


6. A = 10057 / 8. 


A - 7712 7 


C = 333' 


B = 4330 ; 


b = 19.8 


c = 15 


6. A = 4626 9. 


B - 1322' 


C = 624' 


C - 28ir 


6 = 7.43 


a = 7420 


11. 4.93 miles 




130. Page 265 




3. A = 5226' 6. 


A - 4442' 


B = 5923' 


B = 4937' 


C - 6812' 


C - 8540' 


4. A = 4912' 6. 


A = 8342' 


B =. 3736' 


B = 5922' 


C = 93 12' 


C - 3656' 


131. Page 266 




(r) 1.571 (d) 3.14 


(e) 2.09 


(/i) 3.49 (i) 52.4 




(d) 210 (e) 1200 


(/) 176.4 ( 


(c) 0.00000485 (e) 


3.152 


(d) 0.1778 (/) 


5.24 


l15' (c) 9140 ; 


(d) 34346' 



393 



133. Page 271 



3. Each side = STT in. 

6. 3000 miles, 3638 miles, 2750^ miles 

8. (a) c = 30, a = 90, 6 - 90 



394 PLANE AND SPHERICAL TRIGONOMETRY 

135. Pages 276 to 277 



1. (a) c - cos- 1 -2 3. (a) A - tan- 1 2 

4 

(b) B sec- 1 -\/3 (b) Impossible 

(c) c tan" 1 2 (c) a = tan~ l f 

(d) A - sec- 1 4 (d) c = ir - sec~ 

(e) b - tan- 1 ^/| (e) A - cos" 1 f 

(f) Impossible (/) B =- sec" 1 \/3 
8. (a) cos c = cot A cot B 

137. Pages 280, 281 

1. b - 214'5", c = 1045'55", A - 789'22" 

2. a - 4443'49", b - 1459'33", A = 7521'53" 

3. b - 1049'17", c = 11820'20", A = 9555'2" 

4. A - 5216 / 26 // , B = 5726'33", b = 477'32" 
6. a - 5821'28", A 6511 / 30 // , B = 536'40" 

6. b = 2737'26 // , B = 6842 / ll", A - 15548'0" 

7. a = 1274'30", b => 500'0", A - 1203 / 60 // 

8. a = 2215'43", b - 2424 / 19", B - 508'21" 

9. o - 11959'46", b - 12010'3 / ', c - 7526 / 58" 

10. a - 500'0", b - 5650 / 49 // , B = 6326'4 // 

11. b - 5153 / , A = 2728'38 // , B = 7327'11 // 

12. c = 5420', A - 4659'43", B = 57 59'19" 

13. b - 15527 / 54", c = 1429 / 13", A - 541'16" 

14. c = 133 32'26", A = 12640'24 // , B = 4713'43 // 

15. c 5420', B - 4659'43 // , A - 57 59'19" 

16. a - 500'4", b = 1435 / 12 // , c = 12055 / 34" 

17. a - 6733'27", b = 10045', c = 94 5' 

18. a - 5153', B - 2728 / 38 // , A 7327 / ll // 

19. b - 9621'59 // , c = SG^S'O", A = HS^l'lS" 

20. a - 4959'58", c - 9147'40 // , B = 92 8'23" 

22. Z> 690.98 miles, L 2 = 3931'18' / , C = 8019 / 23" 
24. B = 5348'27 // 

138. Page 282 

1. ai - 6950'24 / ', ci - 7345 / 15", Ai - 7754 / 
a 2 - HO^'SG", c 2 = 10614'45", A 2 - 1026' 

2. ai = 1864'38", Ci - 1272'27", Ai - 2357'19" 

a 2 = 1615'22", c 2 - 5257'33", A 2 = 1B62'41" , 

3. ai - 2559'28", c t = 3320 / 13 // , Ai - 5253'0" 
o 2 - 1540 / 32", c 2 - 14639'47", A 2 - 1277 / 0" 

4. bj - 2814'31 X/ , ci = 7853'20 // , Bi - 2849'57" 
b 2 = 15145 / 29 // , c 2 = 1016 / 40 // , B 2 - 15110'3 // 

6. bx =- 394 / 51", ci - 13650 / 23", Bi - 679 / 43" 
b 2 - 14055 / 9 // , 02 - 439'37 // , B 2 - 11250'17" 



ANSWERS 395 

6. 01 - 6036'10", ci - 6842'59", Ai - 6913'47" 
a 2 - 11923'50", c 2 - 111 !?'!", A 2 - 11046'13" 

139. Page 284 

1. (a) a' - 440.9', 6' - 7949.9', c' - 8116.7', C" = 90, A f - 4440'; 
B' = 8128.5' 

2. (a) sin A' sin C' sin a' 

3. (6) a' = 1339.7', B' - 10818.3', c' - 7335.3' 

140. Page 286 

1.0- 6836'13", 6 5919'4", C = 10326'36" 

2. a - 6746'12", 6 - 7821 / 32", B = 7724'34" 

3. 6 = 11745'28", A - 9627 / l // , C = 930'51" 

4. a 9422'46", 6 = 6948'42", (7 = 8823'11" 

5. a - 10656'53", B = 849'46", C = 28 3 ; 4 /; 

6. A 4 10521'16", B - 160 13'48", C = 10425'45" 

141. Pages 285 to 288 

1. (a) c 6632 / 6", A - 4155 / 45", B = 7019 / 15 // 
(6) a - 10453'1", fc - 13339 / 48 // , C = 10441 / 37" 

(c) a = 5441 / 35 // , 6 = 10421 / 28 // , c = 98 14'24" 

(d) a! = 2011 / 16", c, = 12916 / 38 // , Ai - 2628 / 31 // 
a 2 - 15948 / 44 // , c 2 = SO^S^", A 2 = ISS^l^" 

(e) b - 8517'16", A - 1735 / 57 // , C = 104 31'13" 
(/) Impossible 

2. ( ) a - 6 = 3245 ; 6", (7 - 105 49'32" 
(6) c = 4615'12 // , a - 6 = 11232'20" 

3. 6020'66" 

6. Ci - 65 22'31", (7 2 - 11437'29", 61 - 13024'35 // , 6 2 - 7735 / 39" 
Bi = 13520'37", B 2 = 6421'40" 

7. 247.95 miles 9. 856'31", 856'44 // 

10. L - 3955'24 ;/ N, X - 6053'17" W, C - 98297" 

11. L - 248'22 // N, 7) - 3067.7 miles 

12. L - 538'42" N, X - 17649'56" W 

13. 1973.9 nautical miles 14. L = 5517'42" N, X = 180 

142. Pages 290 to 291 

3. (a) A = 7123'00" 4. (a) b = 4413'45 // 

(6) B - 5337'47" (b) B - 13118 ; 

144. Pages 294, 296 

1. (a) a - 4220 / 12" 2. (a) 13740 ; 3. A - 23*11' W 

(b) a - 64 10'34" t (6) 7949' 

(c) a - 10010'68" 

7. (a) B - 11435'50", C = 31 ^OW' 

(b) B = 4252'8 // , C - 2845'18 // 

(c) B - 21 3'6" f C - 266 ; 0" 



396 PLANE AND SPHERICAL TRIGONOMETRY 

8. (a) A' = 13739'48", V = 6524'10", c' = 14820'5" 

(b) A' - 11549'26", 6' = 1377'52", c' = 15114'42" 

(c) A' - 7949'2", 6' - 15856'54", c' = 15354' 

147. Pages 299, 300 

2. (a) A - 3311'20", J5 = 5043'44", C = 1083r52" 

(b) A = 3446'44", B = 816'4", C = 816'4" 

(c) A = 14513'20", B = 9854'0", C = 816'4" 

(d) a - 769'49", b = 12733'10", = 769'49" 

(e) a = 816'0", b = 3446'42", c = 9853 / 56" 

(/) a - 14648'40", b = 7128'8 // , c = 12916 / 1G // 

3. (a) A 11844'10", J5 == 2938'9", ^7 = 687'32" 

(b) A = 12353'48", B = 5746 / 56 // , C = 4651'50" 

(c) A = 8152 / 32", B = 9731'5 // , C = 1113'42" 

(d) A = 3459'19", B = 15013'15", C = 33ll / 39" 

(e) a - SG^l^S", ft = 12657'52", c = 13921 / 22" 
(/) a = 5117 / 31", b = 64 2'47", c = 5ll7'31" 

(^) a = 9744'19 // , 6 = 5349'25", c = 10425'9" 
(h) a = 11510', 6 = 8418 / 28", c = 319'14 // 

4. (a) a' - 14648'40", 6' = 12916 / 16 // , c' = 7128 / 8" 

149. Page 304 

1. (a) b - 4220 / 12", A - 3139 / 54 // , C = 11435'50 // 

(b) a - 8526'28", B = 14953'42", C = 3754'6 // 

(c) A = 3913 / 54", B - 6326 / 6", c = 15642'58" 

(d) a = 16529 / 53 // , 6 = 154 17'43", C = 9319 / 34" 
(/) a - 50ll / 37", B = 7729'48", c - 153 40'13" 

2. (a) 4928' (6) 6935' (c) 1520' (d) 10419' 

3. (a) a = 5756 / 56", 6 = 13720 / 32 // , C = 9448'13 // 

(5) b - 10047'46", A = 962 / 12", C = 125 43'44" 

(c) c = 10412 / 55", A = 6348'26", B = 51 46'38" 

(d) c = 10839'11", A = 6448'54", B = 4023'16 // 

(e) c - 15618'49", A - 2942'0", fi = 412 / 38" 
(/) a - 2357 / ll", 6 = HS^'IS", C = 1025 / 46" 

4. (a) c = 95'14 // , A - SG^O'O", B = HS^S'SG" 

(6) c = 7341 / 2", A = ISO^S'O", B = 12826'27" 

160. Pages 306, 307 

1. ci - 10419'10 // , Ai - 5219'33", (?i - 12442 / 2 // 
c 2 - 1810 / 14", A 2 - 12740 / 27", C 2 = 1520 / 32" 

2. fe 1518'34", c = 3859'34", C = 9840'56" 

3. bi = 55 25'2", d - 8127'26", Ci = 11922 / 28" 

b 2 - 12434'58", c 2 = 16234 / 27 // , C 2 - 1644V55" 

4. bi = 8115 / 15", ci - 11010'50", d - 11943'48" 
b 2 - 9844'45", c 2 - 13845'26", C 2 = 14224'59" 

5. Impossible 

6. c - 8857 / 44 // , A - 5144 / 11 // , B - 13929 / 35 / ' 



ANSWERS 397 



161. Pages 307, 308 

1. A - 12618'42", B - 11942'8", C = 11151'42" 

2. c - 8937'43", A - 2942'0", B - 13857'22" 

3. a - 12334 / 46 // , 6 - 7556'32", c = 1050'18" 

4. 6 = 8812'19", C - 7815'46", a - 15243 / 49 // 
6. a = 11426'50", c = 8233'31", C = 7910'30" 

6. c - 15338'40", A - 2942'34", B - 4237 / 18 // 

7. 01 * 4237'18", d = 12941'5", Ci - 8954'19" 
a 2 - 13722'42", c 2 = 1958'36", C 2 = 2621'18" 

8. A - 5929 / 42 // , B = 6249'42", C = 6550'48" 

9. a = 11030'23", b - 3647'37", C - 13512'15" 
10. a = Sl^T'Sl", 6 = 64 2'47", c - S 



154. Page 312 

1. c - 13549'19", 6 = 14637 / 15", A = 1058'17" 

2. a - 40l / 5", 6 - SS^l'S", C = 1303'48" 

3. c - 12010'52", A = 6513'4", B - 4927'53" 

4. a - 6934'44", B - 1355'14 // , C - 5029 ; 54" 
6. c = 10412'52 // , B = 5146 / 38' / > A = 6348 / 24 // 

6. 6 - 10047 / 46 // , A = 962 / 12", C - ^S^S^" 

7. c = 10839'11", B - 4023 / 17 // , A = 64 48 ; 55" 

8. a = 6528'34 // , B - 14814 / 43 // , C = 449'3" 

9. o - 14524'53 // , b - 13945'58", C - 4946'16" 
10. a - 2357'9", c = HS^'IS", B - 1025'52" 

155. Pages 314, 315 

1. c - 12010 / 52 // , A - eS ^^", B = 49 27'53" 

2. a - 6934'44 // , B - 1355'14", C - 5Q29'54" 

3. c = 10412 / 52", B - 5146'38", A - 6348 / 24" 

4. 6 - 10047 / 46", A - 962'12", C - 12543'46" 
6. c - 10839'11", B - 4023'17 // , A - 6448 / 55 // 

6. a - 6528'34", B - 14814 / 43 // , C =- 449 / 3" 

7. a - 14524'63", b = 13945 / 58 // , C - 4956 / 16" 

8. a 2357 / 9 // , c - 1182'15", B - 102 5'52" 

10. c - 13549'19", b - 14637 / 15", A = 1058'17" 

11. a - 40 1'5", 6 = 38 31'5", C - 1303'48" 

156. Page 316 

l.o- 112 10'4" 3. c - 8857'41" 

2. c - 7341'0" 4. c - 373 / 52 // 

5. A - 5144'7", B = 13929 / 36 // 

158. Page 319 

1. Bi - 4237 / 30 // , Ci - 1601'43", Ci - 16339'4" 
B 2 - 13722'30 // , C 2 = 5019'3", c 2 - 905 / 18" 

2. B - 13125'11", C - 10818 / 66 // f c - 7821'6" 

3. Bi = 12047'28", Ci - 9742'38", d = 6#41 / 57 // 
B 2 - 59 12'18 /; , C 2 - 29 9'0", c 2 - 2357'27 // 



398 PLANE AND SPHERICAL TRIGONOMETRY 

4. Ci = 5924'20", Bi = 11540'1", 61 = 9733'11" 

C 2 - 12035'40", 2 = 2659 / 51", 6 2 - 2957'19" 
6(a). 6 - 7647'13", a - 9646 / 12", A - 9924'13" 
6(6). &! - 10949'57", d = 9821'33", Ci = 10955'11" 

62 = 7010 / 3", c 2 = 16848'53", C z = 16922'45" 
6(a). c, = 12056'49", 61 = 4818'43", 1 = 5855'29" 

c 2 - 593'11", 6 2 = 1208'55", B 2 = 9721 / 31" 
6(6). 61 = 590 / 17", ci = 11821'34", Ci = 9512'4" 

6 2 - 120W43", c 2 = 4352 / 14 // , C Y 2 = 5139 / 22" 

169. Page 320 

1. A = 6833'42", B = 13048'18", C = 940 / 48" 

3. Impossible. 

4. a = IGS^'G", 6 = IGS^O^", c = ll^'G" 

6. A - 6549'48", B - 5632 / 48", C - 11656'48" 
6. No solution. Examine the polar triangle. 

160. Pages 320, 321 

1. A - GS^S'SS", B = 5146'12", c = 10413 / 27 // 

2. B = 9538 ; 4", C = 9726 / 29 // , a - 6423'15" 

3. a = 40l / 5 // , 6 = SS^l'S' 7 , C = ISO^'SO" 

4. B! = 4237'17 // , Ci = IGO !^", c t = 15338'42" 
B 2 = 13722 / 42", C 2 = 50 18'55", c 2 = 905 / 41 // 

6. B - GS^S'lO", C = 9726'29", c = 10049'30 // 

6. 6 - 4152 ; 35", c - 41 ^S'^', C = 6042'46 // 

7. A = ai !^", B =* 8 38'46", C = 1553r36" 

8. a - 8720'28 // , 6 - 76 44'2", c - gS 

9. 4423 / 16" N 

10. L - 2244'22" S, 7 = 166 3 r P: 

11. L = 4254 ; 52" N, 7 = 993 / 30" E 

12. L - 413 / 50" N, 7 - 16819'20" W 

13. C = 224 8'45", D = 5832 mile 

14. A = HO^l'S", J5 = 4856'16 // , C = 



163. Pages 326 to 328 

6. C n - 3113'38", D - 6386.7 miles 

6. C n = 2171 / 18 // 

7. D - 6779.9 miles 

8. C n = 24129'52" 

9. C n - SG^S'lS", D - 5213.7 miles 
L v = 3432'27" N, X v = IGS !^!" W 

10. Cn - 2248'48", D - 5832 miles 

11. L - 4455 / 16" 

12. (a) 439' W (d) 2031'28 // N 

(6) 35 53' N (e) C n = 3166'17" or 21156 / 17", 6988.9 miles 

(c) 3234'36" W (/) 2870.4 miles 

13. Cn = 29742 / 24", C - 22544'48 // , D - 5992.0 miles 



ANfiU'ERfi 



399 



3. /,, = 20812'00" 

h = 5910 / 22" 

4. Z n - 20346'46" 

h = 2142 / 43 // 
6. Zn = 4440'43" 

h = 5139'30" 
6. Z n = 7311'42" 

h = 6413'50 // 



166. Pages 332, 333 

7. Z n = 31214'54" 

h = 3113 / 24" 

8. Z* = 1453'3\" 

h = 35WW 

9. Z n = 12518'40" 

h = 4553'20 /; 

10. Z n = 8559'36" 

h - 3640'18" 



11. h = 2242'25" 

12. h = 6413'52 // 

13. k = 3113 / 25 // 

14. h == 5536'22 // 

15. h = arwso" 

16. h = 5910'15 // 
18. h = 2 ir50 /; 



A = W 6659'30" N at 



167. Page 335 

1. A = E 2928'6" S 2. 4 h 37 m 48 8 A.M. 

3. Summer: sunrise at 4 h 37 m 48 s A.M., sunset at 7 h 22 m 12 s P.M. 
Winter: sunrise at 7 h 22 m 12 8 A.M., sunset at 4 h 37 m 48" P.M. 

4. (a) March 21 : sunrise at 6 h O m O 8 A.M., sunset at 6 h O m (f P.M. 

Deeeniher 21 : Minnse at 10 11 19 m T A.M., sunset at l h 40 m 53 H P.M. 

June 21 : sunrise at l h 4() m 53 s A.M., sunset at I0 h 19" 1 T P.M. 
(6) March 21: A = 0'0" at sunrise; A = O^'O" at sunset 

December 21: A = K G059'30 // S at sunrise; A = W ()(>59 / 3() // S 

at sunset 

June 21: .1 = K W>59'30" X at sunrise; 

sunset 
(c) Length of longest day: 20 h 38'" 14" 

Length of shortest day: 3 h 21 m 4(i 8 
6. (a) 10N ' (</) 10S 

(6) 10S (c) 30.25ft. 

(c) h = 1327, h = 3327' 

168. Page 337 

2. (a) t = 7 h 8 m 2" A.M., Z n = 7926 / 13 // 

(b) t = 7 h 10 m 4l fl A.M., Z n = 8458'52" 

(c) t = 6 h 50 m 25 8 A.M., Zn = 8131 / 5 // 

3. t = 8 h 23 m 50 M A.M., Z n = l()044 / 48 // 

4. t = 9 h 10 IU 46 8 A.M., Zn = 12546'0" 
6. / = 4 h 37 m 46 s P.M., Z n = 27243 / 40" 
6. / = 3 h 5 m 18" P.M., Z n = 261 6'0" 



1. 60 1^ 

2. 15 h 42 m 30" 



1. 



(a) 16 h 22 m ; (b) 3 h 38 1 
& 48 m 40 8 



17623'\5" W 
129'15" K 
12423 / 45' / W 



169. Pages 339, 340 

5. X 2 = iST, - ,ST 2 H- Xi 

6. I8 h 19 m 40 8 

7. 23 h 45 m 22 8 

170. Page 341 

4. X - GO^'O" W 

5. X = lll^'SO" W 

6. X - HG^'IS" W 



400 PLANE AND SPHERICAL TRIGONOMETRY 



171. Page 343 


1. L = 7. 


L 


= 3350' N 


12. L = 3733' N 


2. L = 30 N 8. 


L 


= 1224' S 


13. L - 7422' N 


3. L - 50 X 9. 


L 


= 841' S 


14. L = 3724' S 


4. L - 46' X 10. 


L 


= 


16. L - 45 32' X 


6. L = 7240' S 11. 


L 


= 7ll / X 


16. Impossible 


6. L = 4658' X 








172. Page 344 


1. (a) L! = 1326'28"S 




(b) L t = 


5821'19" S 


L 2 = 6121'31" X 




L 2 = 


4222'21" X 


2. (a) L, = 2541'32" N 




(c) Lt = 


1015'58" X 


Zi = 2550'0" 




L-2 = 


2458'58" N 


L 2 = 841'32" X 




Zi = 


7729'28" 


Z 2 = 2850'0" 




Z< 2 = 


10230'32" 


(6) Li = 1307'20" S 




W) ^ = 


4422'51" X 


L 2 = 7255'50" N 




Z = 


1704'0" 


Zi = 32133'20" 








Z> = 21826'40" 









173. Pages 344 to 349 

2. Z n = 23753'17" 

3. h = 1348'1", Z n = 12526'9" 

4. Li = 2653'48" N, L* = 7119'0" X, Z, = X 450'0" W, 
Z 2 = N 1350'0" W 

6. L! = 2542'1" S, L, = 84ri" S, Z, = S 1050'0" K, 
#, = S 750 / 0" E 

6. (a) /., - 314'46" S, /,* = 4323 / 10" S, Z, = S 25 tJ l5 / 29" K, 

Z, = S 15444'31" K 

(b) Li - 1129'32" S, L 2 - 0239'40" X, Z, = X 41r54" K, 
Z 2 = N 13858'5" E 

7. (a) < = 4 h 27 m 4fi s P.M., Z n = 27243'40" 
(6) < = 10 h 7 m 44 s A.M., Z n = 3456'36" 

8. Comes within 7.6 nautical miles of the Chicago position 

9. 1) - 3355.2 miles, C n = 8648'48" 

10. I) = 6748.6 miles, C n = 824 / 28", L, = 2829'44" S, 
X, == 13613 / 45" E 

11. 1) = 4461.7 miles, C n = 30213'45" 

12. D - 6430.6 miles, Cn = SOO^O^" 

13. L = 4325'37" X, 1329.5 miles north of Honolulu 

14. 169 7'4" W 

16. L = 6610'2" X, X = 16734 / 16" E 

16. (a) L = 5721'21" N, X = 1733'33" W 
(6) L - 4437'18" N, X = 6820'35" W 

17. 15223' 19. d - 3240'36" 8 

18. 9957 / 30" 20. 3 h 26 ni O 8 E 
21. 5545' N 



ANSWERS 401 

22. (a) 4 h 50 m 59" A.M., 7 h 9 m 1" P.M 
(6) 5 h 47'" 50 s A.M., f> h 12 m 4" P.M. 

(c) 5 h 50 m A.M., (> h IT, 1 " P.M. 

(d) 6 h 12 ni A.M., 5 h 48"' P.M. 

23. (a) 18 h 28 m 21 s ; (b) 5 h 31 m 3(>" 

24. / = 4 h 2<T 1 9" K, A = K 3335'3" X 

26. (a) 2 h 4 m 28 s , 5 h f> m 40", 1 4 h 44 ltl 25", 2 h 4 m 28" 
(6) l h 41 in 5", 11 h 22 rn 15 s , 9 h 15 m 35 s , 1 h 41'" 5 s 
(c) l h 33'" 42 s , 8 h 52 m 37 s , 12 h O 111 0", l h 33 m 42 s 

26. (a) 4058' X (c) 194l)' S (c; 4^6' N 
(6) 4142' X (d) 7240' S (/) 930' S 

27. For visible louor culniinatioii, L, f/, and braniitf must all })i* of thu same 
name, with L -h '/ > 90 and at a lower culmination /? < <A 

28. (a) 3830' N (c) 74^22' X 
(fc) 7563' S (d) 37"24' S 

29. (a) 7 h 43 rn 15" (r) S 5714'39" K 
(h) T>.91 

30. 3 h 59 rn 23" P.M. 32. (n) 93 C 19'15" K 

31. 2 h 58 m 44 8 P M (b) 92'27 // K 

33. The shadow stretches from foot of polo S 7122' W 

34. / = 75ir 37. f h 58 m \ M., 5 h 2 rn P.M. 
36. 13.8 ft. 38. 89.7 miles, 341 36 miles 
36. 120 39. 17 14'40" 



FIVE-PLACE LOGARITHMIC AND 
TRIGONOMETRIC TABLES 



BOOKS BY 

LYMAN M. KELLS, WILLIS F. KERN, 
and JVMES R. BLAND 

PLANK AND SPHERICAL TRIGONOMETRY 
Second Edition 
6x9, Illustrated. 
With tables, 510 pages. 
Without tables, 401 pages. 

PLANE TRIGONOMETRY 
Second Edition 
6x9, Illustrated. 
With tables, 418 pages. 
Without tables, 303 pages. 

LOGARITHMIC AND TRIGONOMETRIC TABLES 
118 pages, 6x9. 



FIVE-PLACE LOGARITHMIC 

AND 

TRIGONOMETRIC TABLES 



BY 

LYMAN M. KELLS, PH.D. 

Associate Professor of Mathematics 

WILLIS F. KERN 

Assistant Professor of Mathematics 
AND 

JAMES R. BLAND 

Assistant Professor of Mathematics 
All at the United States A'araZ Academy 



FIRST EDITION 
FIFTEENTH IMPRESSION 



McGRAW-HILL BOOK COMPANY, INC. 

NEW YORK AND LONDON 
1935 



COPYRIGHT, 1935, BY THE 
McGRAw-HiLL BOOK COMPANY, INC. 



PRINTED IN THE UNITED STATES OF AMERICA 

All rights reserved. This book, or 

parts thereof, may not be reproduced 

in any form without permission of 

the publishers. 



THE MAPLE PRESS COMPANY, YORK, PA. 



PREFACE 

A table of logarithms should be accurate, it should be easy to under- 
stand, and it should be as easy to use as possible. The authors, in the 
tables offered here, have attempted to make improvements along these 
three lines. 

The tables used in trigonometry and its applications have been 
checked many times and have been carefully read against other tables. 
If, in spite of this thoroughness in compilation, errors are discovered, 
the authors would appreciate having them pointed out. 

Frequently students fail to understand the process of linear interpola- 
tion. It is explained in this book by means of a simple diagram which 
gives the idea almost at a glance. 

The table of logarithms of trigonometric functions (Table II), the 
most important one for trigonometry, has a number of new features. 
The proportional parts are tabulated for each second from 0" to 60", and 
bold-faced numbers have been so used as to avoid ambiguity. Whenever 
there is a choice of two numbers one of which is written in bold face, 
tho bold-faced number is always chosen. The simplicity of operation 
introduced by this plan gives a gain both in speed and in accuracy. 
In the table proper all six functions are tabulated, and bold-faced num- 
bers are used in such a way as to enable the user to locate approximate 
position by using them only. It is believed that the gains due to these 
innovations are decidedly worth while. 

LYMAN M. KELLS. 
WILLIS F. KERN. 
JAMES R. BLAND. 
ANNAPOLIS, Mn., 
July, 1935. 



CONTENTS 

1'Ai.fc 

1'llKKAt'K . - . ^ 

TAHLK I 
COMMON LOGARITHMS OF NUMBERS 

Aitr 

1. Introduction 1 

2. Characteristic, and Mantissa 1 

3. To Find the Mantissa. Special Case 'J 
t. Interpolation . 2 
5. To Find the Number Corresponding to a Given logarithm 3 

TAIILE II 
LOGARITHMS OF TRIGONOMETRIC 1 FUNCTIONS 

i) Table of Logarithms of Trigonometric Function* 5 

7 Given the Angle, to Find the Logarithm of a Trigonometric Function 5 

S. Given the logarithm of a Trigonometric Function, to Find the Angle 7 

<) Angles Near and 90 S 

TAHLK III 
TRIGONOMETRIC 1 FUNCTIONS 

10. Table of Natural Values of Trigonometric Function?- 10 

Table 1 Five-place Table of Common Logarithms of Number^ 13 

Table II. Logarithms of Trigonometric Functions 37 

Table III. Natural Trigonometric Functions 91 

Table IV. Radian Measure, to 180, Radius = 1 115 

Table V. Haversines 116 



FIVE-PLACE LOGARITHMIC AND 
TRIGONOMETRIC TABLES 

TABLE I 
COMMON LOGARITHMS OF NUMBERS 

1. Introduction.* The power L to which a given number b must be 
raised to produce a number N is railed the logarithm of N to the base b. 
This relation expressed in symbols is 

b L = N. 

It appears at once that 6 must not be unity and it must not be nega- 
tive. In the following set of tables, 10 is used as base. This system is 
called the common system or the Briggs system. Another important 
system, called the natural system, has e as base, where e = 2.71828 
accurate to six figures. 

2. Characteristic and mantissa. The common logarithm of any real, 
positive number may be written as an integer, positive or negative, plus 
a positive decimal fraction. The integral part is called the characteristic 
and the decimal part the mantissa. The characteristic may be written 
by using the* following rules: 

Rule 1. The characteristic of the common logarithm of a number greater 
than 1 is obtained by subtracting 1 from the number of digits to the left of 
the decimal point. 

Rule 2. The characteristic of the common logarithm of a positive number 
less than 1 is negative and its magnitude is obtained by adding 1 to the 
number of zeros immediately following the decimal point. 

If the characteristic of a number is n (n positive), it should be 
written in the form (10 n) 10. To obtain directly the logarithm of 
a number less' than 1, subtract from 9 the number of zeros immediately 
following the decimal point y and write the result before the mantissa and 
- 10 after it. 

The method of finding the mantissa of the logarithm of a. number will 
be explained in the succeeding articles. 

* Sinoe the theory of logarithms is treated completely in algebra rind in trigo- 
nometry, only the actual manipulation of the tables is explained heie. 

1 



2 EXPLANATION OF TABLE I 

EXERCISES 

Verify the characteristic of the logarithm of each of the numbers N written below. 
N log AT N lo K N 

1. 6.830 0.83442. 8. 58.73 1.76886. 

2. 68.30 1.83442. 9. 0.6740 9.82866 - 10. 

3. 6830 3.83442. 10. 0.007500 7.87506 - 10. 

4. 683,000 5.83442. 11. 6.870 X 10 5 5.83696. 

5. 0.7860 9.89542 - 10. 12. 5.860 X 10~ 4 6.76790 - 10. 

6. 0.007860 7.89542 - 10. 13. 3.990 X lO" 6 4.60097 - 10. 

7. 0.0007860 6.89542 - 10. 14. 7.330 X 10 2 2.86510. 

3. To find the mantissa. Special case. The mantissa, or decimal 
part of the logarithm of a number, depends only on the sequence of the 
digits and not on the position of the decimal point. 'Fable I lists the 
mantissas, accurate to five decimal places, of the logarithms of all integers 
from 1 to 10,000. 

The change in the mantissas of the logarithms is so slow that the 
first two figures do not change for several lines of the table. Conse- 
quently the appropriate first two figures are printed in the first column 
before the first full row to which they apply. Also the appropriate first 
two figures appear at the left of the first line of mantissas on each page 
An asterisk in any row indicates that the first two figures are to be found 
at the left of the next row. 

To find the mantissa of the logarithm of a number locate the first 
three digits of this number in the left-hand column headed N and the 
fourth digit in the row at the top of the page. Then the mantissa of the 
given number containing four significant figures is in the row whose; 
first three figures are the first three significant figures of the given num- 
ber, and in the column headed by the fourth. Thus to find the logarithm 
of 76.64 find 766 in the column headed N, follow the corresponding row 
to the entry in the column headed by 4. This entry 88446 represents 
the mantissa required. Hence we have 

log 76.64 = 1.88446. An*. 

EXERCISES 

Verify the logarithms in the exercise of 2. 

4. Interpolation. When a number contains a fifth significant figure, 
we find the logarithm corresponding to the first four figures as in 3 and 
then add an increment obtained by a process called interpolation. This 
process is based on the assumption that for relatively small changes in 
the number N the changes in log N are proportional to the changes in N. 
The following example will serve to illustrate the process of interpolation. 

The expression tabular difference will be used frequently in what 
follows. The tabular difference, when used in connection with a table, 



EXPLANATION OF TABLE I 3 

means the result of subtracting the lesser of two successive entries from 
the greater. 

Example. Find log 235.47. 

Solution. We first find the logarithms in the following form and 
then compute the difference indicated: 

Iog235.40\ | = 2.37181K) 

log 235.47J >10 = ? / >18 (tabular difference*) 

log 235.50 I = 2.37199 ) 

By the principle of proportional parts, we have 

To = !' or d = iV 18) = 12 ' 6 = 13 < llPHrl y>- 

Adding 0.00013 to 2.37181, we obtain 

log 235.47 = 2.37194. Am. 

The increment 12.6 was rounded off to 13 because we are not justified 
in writing more than five decimal places in the mantissa. 

The essence of this procedure is embodied in the following statement. 
To find the logarithm of a number composed of five significant figures, 
fiist find the logarithm corresponding to the first four figures and to it 
add one-tenth of the tabular difference multiplied by the fifth digit. 

To shorten the process of interpolation, 10 5 times each tabular differ- 
ence occurring in the table has been multiplied by 0.1, 0.2, . . . 0.9, 
and the results have been tabulated on the right-hand sUes of the pages 
on which these differences occur. The abbreviation Prop. Farts written 
at the top of the page over these small tables abbreviates the words 
proportional parts. To interpolate in the example just solved, locate the 
Prop. Parts table headed 18 and find opposite 7 in its left-hand column 
the entry 12.6 ( = 13 nearly). In general, this difference should not be 
computed but should be obtained from the number opposite the fifth 
digit in the appropriate table of proportional parts. 

EXERCISES 

Verify the following logarithms: 

1. log 7012.6 = 3.84588 8. log 0.056321 = 8.75067 - 10. 

2. log 54.725 = 1.73819. 9. log 4,574,000 = 6.66030. 

3. log 0.87364 - 9.94133 - 10. 10. log 568.91 = 2.75504. 

4. log 3.7245 = 0.57107. 11. log 4.3965 X 10 5 = 5.64311. 
6. log 0.00065931 = 6.81909. IO 12. log 10.905 = 1.03763. 

6. log 25.819 = 1.41194. 13. log 0.0025725 = 7.41036. * lO 

7. log 2.3454 = 0.37022. 14. log 0.000032026 = 5.50550 - 10. 

5. To find the number corresponding to a given logarithm. If log 

N = L, the number N is called the antilogarithm of L, The sequence of 
* For convenience the decimal point has been omitted. 



4 EXPLANATION OF TABLE I 

digits of a number N corresponding to a given logarithm L is found irom 
its mantissa, and the decimal point is then placed in accordance with 
the rules of 2. 

Example. Given log N = 1.60334, find N. 

Solution. The mantissa .60334 lies between the entries .60325 and 
.60336 of Table I. Using the table and computing the differences 
indicated, we write the following form: 

1.60325) ) = log 40.110 
1.60334/ y >ll = log AT 
1.60336 ) = log 40.120 

Assuming that changes in the logarithm are proportional to the corre- 
sponding changes in the number, we write 




IT = w or x = 10 = 8 



= 10 (n) = 



Hence 

N = 40.118. An*. 

The essence of the process of interpolation is indicated in the fore- 
going procedure. However, in practice, the student should always 
interpolate by using the table of proportional parts. The fifth figure 8 
should have been obtained from the table of proportional parts. In the 
small Prop. Parts table corresponding to the tabular difference 11, we 
read the fifth figure 8 in the left-hand column opposite the entry 8.8, 
the entry nearest to 9. 

EXERCISES 

Verify the following antilogarithms : 

1. 3.57351 = log 3745.'5. 8. 4.76224 = log 57842. 

2. 2.82315 = log 665.50. 9. 6.51738 - 10 = log 0.00032914. 

3. 0.12112 = log 1.3217. 10. 1.49715 = log 31.416. 

4. 1.92594 = log 84.321. 11. 4.21691 - 10 = log 16478. 
6. 9.47954 - 10 = log 0.30167. 12. 5.09873 = log 125520. 

6. 8.65636 - 10 = log 0.045327. 13. 9.27951 - 10 -= log 0.19033. 

7. 0.37976 = log 2.3975. 14. 7.88000 - 10 = log 0.0075858. 



TABLK II 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 

6. Table of logarithms of trigonometric functions. Table II gives 
the logarithms of the sines, cosines, tangents, cotangents, secants, and 
cosecants of angles at intervals of 1' from to 90. The names of the 
functions written at the top of any page apply to angles having the 
number of degrees written at the top of the page, and the function names 
written at the bottom apply to angles having the 'number of degrees 
written at the bottom. The left-hand or the right-hand minute column 
applies according as the dumber of degrees in the angle is written on the 
left side or on the right side of the block of numbers under consideration. 

For example, to find log sin 32 46', we find the page at the top of 
which 32 appears, find the row containing 46 in the left-hand minute 
column, and read 73337 in this row and in the column headed I sin. 
Hence log sin 32 46' = 9.73337 - 10. The number 9 was found at the 
head of the I sin column and the number 10 is to be applied to every 
logarithm in the table. Again, to find log tan 142 36', find the page 
at the top of which 142 appears, find the row containing 36 in the right- 
hand minute column, and read 88341 in this row and in the column 
headed I tan. Hence log tan 142 36' = (-) 9.88341 - 10. The minus 
sign in parentheses before the log indicates that a negative number is 
under consideration. The characteristic was obtained as in the first 
example. 

EXERCISES 

Verify the following: 

1. log sin 37 27' = 9.78395 - 10. 

2. log tan 36 41' = 9.87211 - 10. 

3. log cot 28 16' = 0.26946. 

4. log cos 62 20' = 9.66682 - 10. 
6. log esc 69 54' = 0.02729. 

6. log sin 131 10' = 9.87668 - 10. 

7. log tan 142 27' = (-) 9.88577 - 10. 

8. log sec 134 47' = (-) 0.15216. 

9. log cos 45 47' = 9.84347 - 10. 

10. log esc 135 13' - (-) 0.15216. 

11. log cot 132 0' = (-) 9.95444 - 10. 

7. Given the angle, to find the logarithm of a trigonometric function. 
The principles involved here are the same as those involved in finding 

5 



6 EXPLANATION OF TABLE 11 

logarithms and antilogarithms of numbers. Interpolation for seconds 
is accomplished by direct interpolation or by using the columns headed 
d 1' arid the columns headed proportional parts. The following example 
will illustrate the procedure. 

Example. Find log tan 65 42' 17". 

Solution. Using the table to find logarithms and computing differ- 
ences, we write the following form: 

log tan 65 42' 00"\ .) = 0.34533\ 
log tan 65 42' 17"/ >60" ? f* 

log tan 65 43' 00" ) = 0.34566 

Hence assuming that, for small changes, change of logarithm is propor- 
tional to change of angle, we have 

33 60 v^v ' 

Therefore 

log tan 65 42' 17" = 0.34533 + 0.00009 = 0.34542. An*. 

The essence of the process of interpolation is indicated in the fore- 
going procedure. However, in practice, the student should always 
interpolate by using the columns headed d 1' and the proportional parts 
column. 

Each entry in the column headed d 1' gives the difference of the loga- 
rithms between which it is spaced in each of the adjacent columns. In 
each column headed by proportional parts appears - \>, <f<>, -,% . . oi 
the number heading the column. Hence the difference 9 to be applied 
in the case of the foregoing example is found in the proportional parts 
column headed by 33 (the tabular difference for 1' written between 0.34533 
and 0.34566) and in the row with the 17 of the seconds column. Again, 
to find log cot 10 28' 36", we find the entry 73345 for log cot 10 28', no(<> 
the appropriate number 71 in the adjacent column headed d 1', enter 
the proportional parts column headed by 71, read in this column 43 
opposite the 36 of the seconds column; subtract 43 from 73345, and write 
log cot 10 28' 36" = 0.73302. 

It is worthy of note that the changes of logarithms due to the seconds 
of an angle must be added or subtracted according as the value of the 
function for angles near the one under consideration is increasing or 
decreasing with increasing angle. 

EXERCISES 

Verify the following: 

1. log sin 35 17' 8" = 9.76166 - 10. 

2. log cos 48 24' 21" = 9.82207 - 10. 

3. log sec 142 37' 15" = (-) 0.09984. 



EXPLANATION OF TABLE 11 7 

4. log esc 66 21' 57" = 0.07956. 
6. log cot 23 16' 50" = 0.36626. 

6. log esc 128 47' 52" = 0.10826. 

7. log tan - 69 38' 54" = (-) 0.43070. 

8. log sin 197 36' 57" - 9.48092 - 10. 

9. log sin 137 45'i22" - 9.82756 - 10. 

10. log cos 137 45' 22" = (-) 9.86940 - 10. 

11. log sin 209 32' 50" = 9.69297 - 10. 

12. log cos 330 27' 10" = 9.93949 - 10. 

8. Given the logarithm of a trigonometric function, to find the angle. 

The following example will indicate the procedure necessary to find the 
angle when the logarithm of a trigonometric function of the angle is 
given : 

Example. Find if log cos is 9.85391 - 10. 

Solution. Using the table to find logarithms and computing differ- 
ences, we write the following form : 

log cos 44 24' 00") ) = 9.85399 {) 
log cos 44 24' r'f x >W = 9.85391/ >13 
log cos 44 25' 00" ) = 9.85386 ) 
Hence 

go = 15' or x = S (60) = 37 " (nearly) ' 

and 

B = 44 24' 37". Am. 

The essence of the process of interpolation is indicated in the fore- 
going procedure. In practice, however, the columns headed d 1' and 
the proportional parts columns should be used in interpolation. Thus, 
to find B in the example just considered, we first find 44 24' and difference 
8 as above, then read 13 in the column headed d V adjacent to and slightly 
below the entry 85399, enter the corresponding proportional parts 
column, opposite the bold-faced one of the five 8's tabulated read 37" 
in the seconds column, and then write B = 44 24' 37". 

When finding the number of seconds in an angle corresponding to a 
given logarithm of a trigonometric function, the student may find several 
identical entries in the proportional parts column involved. In this 
case, and in any case where there is a choice between two or more entries one 
of which is printed in bold face, always give preference to the bold-faced 
entry. 

EXERCISES 

Find the value of 6 less than 360 in the following: 

1. log sin = 9.96162 - 10. Ant. 66 16' 0" and 113 44' 0". 

2. log cos 8 = 9.99537 - 10. Ana. 8 21' 0" and 351 39' 0". 

3. log cot = 0.52368. Ana. 16 40' 13" and 196 40' 13". 



EXPLANATION OF TABLE 11 



4. log tan 6 = 9.50368 - 10. 

6. log cos = 9.96301 - 10. 

6. log sin = 9.84963 - 10. 

7. log cot e = 9.50064 - 10. 

8. log tan = 0.96236. 

9. log see 6 = 0.12358. 
10. log esc = 0.71238. 



Ans. 17 4l 7 18" and 197 41' 18". 
Ans. 23 18' 48" and 336 41' 12". 
Ans. 45 1' 9" and 134 58' 51". 
Ans. 72 25' 38" and 252 25' 38". 
Ans. 83 46' 34" and 263 46' 34". 
Ans. 41 12' 22" and 318 47' 38". 
Ans. 11 10' 53" and 168 49' 7". 



9. Angles near and 90. When angles are near or near 90, 
interpolation based on the assumption of proportional change in angle 
and logarithm may give results considerably in error. For this reason 
it is convenient to introduce the- functions S and T defined by the equa- 
tions S = a/sin a and T = a/tan a. The relative change of the func- 
tions S and T with respect to a is very small when a is less than 3 and, 
as a consequence, the required accuracy of the results is obtained by 
using them. On the first three pages of Table II the columns headed 
log AS* and log T give the common logarithms of S and T 7 , respectively. 

The following formulas apply when the angle involved is less than 3: 
1. For angles less in magnitude than 3. 



(a) log sin a = log a"f log S. 
(6) log tan a = log a" - log T. 

(c) log cot a = colog a" + log T 7 , 

= colog tan a. 

(d) log esc a = colog a" + log S. 
2. For angles a such that 90 - 



(e) log a" = log sin a + log S. 

(/) log a" = log tan a + log T. 

(0) log a 7 ' = colog cot a + log T. 

(h) log a" = crolog csc a + log S. 

*t is less in magnitude than 3. 



(0 log cos a = log (90 - a)" - log 8. 
tf) log cot a = log (90 - a)" - log T. 
(ifc) log tan a = colog (90 - a)" + log T, 

= colog cot a. 

(I) log sec a = colog (90 - a)" + log 8. 
(m) log (90 - a)" = log cos a + log S. 
(TO) log (90 - a)" = log cot a + log T. 
(o) log (90 -a)" = colog tan a + log T. 
(p) log (90 - a)" = colog sec a + log S. 

To find e when log sin 6 = 8.46932 - 10, we first find in the column 
headed I sin the entry nearest to 8.46932, namely, 8.46799. On one 
side of 8.46799 we read log S = 5.31449, and on the other 1 41' = 6060". 
Hence, using formula (e), we write log a = 8.46932 - 10 + 5.31449 = 

* The function log S is often written cpl S, and the function log T, is written 
cplT. 

t The symbol log a" means in thia connection the logarithm of the number of 

seconds in the angle. 

(90 __ a \'f 

t Since cos a = sin (90 a), in this case = > /Qno - ? 

sin ^yu ~ d) 



EXPLANATION OF TABLE II 9 

3.78381. Therefore a = 6078.7". Since 1 41' = 6060", 6078.7" = 
1 41' 19". 

EXERCISES 

Verify the following: 

1. log sin 44' 13" = 8.10930 - 10. 6. log cot 89 3' 11" = 8.21824 - 10. 

2. log cos 89 21' 31" = 8.04899 - 10. 7. log cos 88 41' 20" = 8.35948 - 10. 

3. log tan 32' 23" = 7.97406 - 10. 8. log sin 59' 8" = 8.23554 - 10. 

4. log cot 25' 56" = 2.12241. 9. log tan 1 29' 10" = 8.41403 - 10. 

5. log tan r 10' 9" = 8.30981 - 10. 10. log sec 88 16' 10" = 1.52000. 
Verify the following: 

11. log cos - 8.32967 - 10; $ = 88 46' 33" and 271 13' 27". 

12. log tan $ = 8.11584 - 10; B = 44' 53" and 180 44' 53". 

13. log sin = 8.23468 - 10; 6 = 59' 1" and 179 0' 59". 



TABLE 111 
NATURAL TRIGONOMETRIC FUNCTIONS 

10. Table of natural values of trigonometric functions. Table 11 i 
contains the numerical values of the sines, cosines, tangents, and 
cotangents of angles from to 90 at intervals of 1'. In the case 
of an angle in the range from to 45, the number of degrees in 
the angle and the names of the functions are found at the top of the 
page and the left-hand minute column applies; in the case of angles in 
the range from 45 to 90, the number of degrees in the angle and the 
names of the functions are found at the bottom of the page and the 
right-hand minute column applies. Interpolation must be carried out 
without the aid of difference columns or tables of proportional parts. 

The following examples illustrate the method of using the tables. 

Example 1. Find sin 68 28'. 

Solution. We first find the page at the bottom of which 68 appears 
and then find the row of the 68 block containing 28' in the right-hand 
minute column. In this row and in the column having sin at its foot 
we find 020 to which we must prefix 0.93 to obtain sin 68 28' = 0.93020. 

Example 2. Find sin 38 38' 27". 

Solution. Using the tables and computing differences, we find the 
values exhibited in the following form: 



sin 38 38' 00") ) = 0.62433 

sin 38 38' 27" } 60" = ? 

sin 38 39' 00" ) = 0.62456 

Hence 

x 27 



23 - 60' r * = (oO j 28 = 10 (nearly) ' 
Therefore 

sin 38 38' 27" = 0.62433 + 0.00010 = 0.62443. Ar?s. 

Example 3. If cot 6 = 0.37806, find 8. 

Solution. Using the tables and computing differences, we find the 
values exhibited in the following form: 



17' 00"\ ) = 0.37820) ) 
? r>60 = 0.37806J >33 

18' 00" 1 = 0.37787 ) 



cot 69 

cot ? D60 = 0.37806f">33 

cot 69 

10 



EXPLANATION OF TABLE HI 1 1 

Hence 

^ = 1*, ()r jc = i|(60) = 2,5" (nearly), and 6 = 69 17' 25". /Ins. 

Since cot 6 is positive in the third quadrant, we may also write an 
answer 180 + 69 17' 25" = 249 17' 25". Ans. 

EXERCISES 

Verify the following: 

1. sin 53 42' 0" = 0.80593 5. cos 83 17' 38" = 11678. 

2. cos 31 53' 9" = 0.84911. 6. sin 87 37' 25" = 0.99914 

3. tan 156 42' 13" = -0.43059. 7. cot 13 14' 52" = 4.2475. 

4. cot 27 51' 17" = 1.8923 8. tan 83 40' 30" = 9.0218. 

Find the values of less than 360 D in the following: 

9. sin 6 = 0.89742 Ans. 63 49' 12" and 116 10' 48" 

10. cos = 0.43750. Am. 64 3' 20" and 295 56' 40". 

11. Ian e = -0.92834 Ans. 137 7' 41" and 317 7' 41". 

12. cot e = 1.8923. Ans. 27 51' 17" and 207 51' 17" 

13. cos e - 95140. .Us. 17 56' 14" and 342 3' 46" 

14. sin 6=0 13552. A//.S. 7 47' 19" and 172 12' 41". 



TABLE I 
FIVE-PLACE TABLE OF COMMON LOGARITHMS OF NUMBERS 

From 1 to 10,000 



TABLE I 
FIVE-PLACE TABLE OF COMMON LOGARITHMS OF NUMBERS 

Krorn 1 to 10,000 



N. 


Log- 


N. 


LOR. 


N. 


Log. 


N. 


IX)R. 


N. 


Log. 








20 


1.30 103 


40 


1 60 206 


60 


1.77 815 


80 


1.90 309 


1 


0.00 000 


21 


1.32 222 


41 


1 61 278 


61 


1.78 533 


81 


1.90 849 


2 


0.30 103 


22 


1.34 242 


42 


1.62 325 


62 


1.79 239 


82 


91 381 


3 


47 712 


23 


1.36 173 


43 


1 63 347 


63 


1 79 934 


83 


.91 908 


4 


0.60 206 


24 


1.38 021 


44 


1.64 345 


61 


1 80 618 


84 


92 428 


5 


0.69 897 


25 


1.39 794 


45 


1 65 321 


65 


1.81 291 


85 


.92 942 


6 


0.77 815 


26 


1.41 497 


46 


1.66 276 


66 


1.81 954 


86 


93 450 


7 


0.84 510 


27 


1.43 136 


47 


1 67 210 


67 


1 82 607 


87 


.93 952 


8 


0.90 309 


28 


1 44 716 


48 


1 68 124 


68 


1 83 251 


88 


94 448 


9 


0.95 424 


29 


1.46 240 


49 


1 69 020 


69 


1 83 885 


89 


1.94 939 


10 


1.00000 


30 


1.47 712 


50 


1.69 897 


70 


1.84 510 


90 


1.95 424 


11 


1.04 139 


31 


1.49 136 


51 


1 70 757 


71 


1.85 126 


91 


1 95 904 


12 


.07 918 


32 


1.60 515 


52 


1.71 600 


72 


1.85 733 


92 


96 379 


13 


11 394 


33 


1.51 851 


53 


1.72 428 


73 


1.86 332 


93 


1.96 848 


14 


.14 613 


34 


1.53 148 


54 


1.73 239 


74 


1.86 923 


94 


1 97 313 


15 


.17609 


35 


1 54 407 


55 


1.74 036 


75 


1.87 506 


95 


.97 772 


16 


.20 412 


36 


1 55 630 


56 


1.74 819 


76 


1 88 081 


96 


98 227 


17 


.23 045 


37 


1 56 820 


57 


1.75 587 


77 


1 88 649 


97 


98 677 


18 


.25 527 


38 


1.57 978 


58 


1.76 343 


78 


1 89 209 


98 


99 123 


19 


27 875 


39 


1 59 106 


59 


1.77085 


79 


1 89 763 


99 


99 564 


20 


1 30 103 


40 


1.60 206 


60 


1.77 815 


80 


1 90 309 


100 


2.00 000 



15 



TABLE I 



0-50 



N. 


LO 


1 


2 


3 


4 


5 




6 


7 


8 


9 




1 
2 
3 




00 000 


30 103 


47 712 


60 206 


69 897 


77 


81$ 


84 510 


90 309 


95 424 


00 000 
30 103 

47 712 


04 139 
32 222 
49 136 


07 918 
34 242 
50 515 


11 394 
36 173 
51 851 


14 613 
38 021 
53 148 


17 609 
39 794 
54 407 


20 
41 
55 


412 
497 
630 


23 045 
43 136 
56 820 


25 527 
44 716 
57 978 


27 875 
46 240 
59 106 


4 
5 
6 


60 206 
69 897 
77 815 


61 278 
70 757 
78 533 


62 325 
71 600 
79 239 


63 347 
72 428 
79 934 


64 345 
73 239 
80 618 


65 321 

74 036 
81 291 


66 
74 
81 


276 
819 
954 


67 210 
75 587 
82 607 


68 124 
76 343 
83 251 


69 020 

77 085 
83 885 


7 
8 
9 

10 

11 
12 
13 


84 510 
90 309 
95 424 


85 126 
90 849 
95 904 


85 733 
91 381 
96 379 


86 332 
91 908 
96 848 


86 923 
92 428 
97 313 


87 506 
92 942 
97 772 


88 
93 
98 


081 
450 
227 


88 649 
93 952 
98 677 


89 209 
94 448 
99 123 


89 763 
94 939 
99 564 


00 000 


00 432 


00 860 


01 284 


01 703 


02 119 


02 


531 


02 938 


03 342 


03 743 


04 139 
07 918 
11 394 


04 532 
08 279 
11 727 


04 922 
08 636 
12 057 


05 308 
08 991 
12 385 


05 690 
09 342 
12 710 


06 070 
09 691 
13 033 


06 
10 
13 


446 
037 
354 


06 819 
10 380 
13 672 


07 188 
10 721 
13 988 


07 555 
11 059 
14 301 


14 
15 
16 


14 613 
17 609 
20 412 


14 922 
17 898 
20 683 


15 229 

18 184 
20 952 


15 534 

18 469 
21 219 


15 836 
18 752 
21 484 


16 137 
19 033 
21 748 


16 
19 
22 


435 
312 
Oil 


16 732 
19 590 
22 272 


17 026 
19 866 
22 531 


17 319 
20 140 
22 789 


17 
18 
19 

20 

21 
22 
23 


23 045 
25 527 
27 875 


23 300 
25 768 
28 103 


23 553 
26 007 
28 330 


23 805 
26 245 
28 556 


24 055 
26 482 
28 780 


24 304 
26 717 
29 003 


24 
26 
29 


551 
951 
226 


24 797 
27 184 
29 447 


25 042 
27 416 
29 667 


25 28$ 
27 646 
29 885 


30 103 


30 320 


30 535 


30 750 


30 963 


31 175 


31 


387 


31 597 


31 806 


32 015 


32 222 
34 242 
36 173 


32 428 
34 439 
36 361 


32 634 
34 635 
36 549 


32 838 
34 830 
36 736 


33 041 
35 025 
36 922 


33 244 
35 218 
37 107 


33 
35 
37 


445 
411 
291 


33 646 
35 603 
37 475 


33 846 
35 793 
37 658 


34 044 
35 984 
37 840 


24 
25 
26 


38 021 
39 794 
41 497 


38 202 
39 967 
41 664 


38 382 
40 140 
41 830 


38 561 
40 312 
41 996 


38 739 
40 483 
42 160 


38 917 
40 654 
42 325 


39 
40 
42 


094 

824 
488 


39 270 
40 993 
42 651 


39 445 
41 162 
42 813 


39 620 
41 330 
42 975 


27 

28 
29 

30 

31 
32 
33 


43 136 
44 716 
46 240 


43 297 
44 871 
46 389 


43 457 
45 025 
46 538 


43 616 
45 179 
46 687 


43 775 
45 332 
46 835 


43 933 

45 484 
46 982 


44 
45 
47 


091 
637 
129 


44 248 
45 788 
47 276 


44 404 
45 939 

47 422 


44 560 
46 090 
47 567 


47 712 


47 857 


48 001 

49 415 
50 786 
52 114 


48 144 


48 287 


48 430 


48 


572 


48 714 


48 855 


48 996 


49 136 
50 515 
51 851 


49 276 
50 651 
51 983 


49 554 
50 920 
52 244 


49 693 
51 055 
52 375 


49 831 
51 188 
52 504 


49 
51 
52 


969 
322 
634 


50 106 
51 455 
52 763 


50 243 

51 587 
52 892 


50 379 
51 720 
53 020 


34 
35 
36 


53 148 
54 407 
55 630 


53 275 
54 531 
55 751 


53 403 
54 654 
55 871 


53 529 
54 777 
55 991 


53 656 
54 900 
56 110 


53 782 
55 023 
56 229 


53 
55 
56 


908 
145 
348 


54 033 
55 267 
56 467 


54 158 
55 388 
56 585 


54 283 
55 509 
56 703 


37 
38 
39 

40 

41 
42 
43 


56 820 
57 978 
59 106 


56 937 
58 092 
59 218 


57 054 
58 206 
59 329 


57 171 
58 320 
59 439 


57 287 
58 433 
59 550 


57 403 
58 546 
59 660 


57 

58 
59 


519 
659 
770 


57 634 
58 771 
59 879 


57 749 
58 883 
59 988 


57 864 
58 995 
60 097 


60 206 


60 314 


60 423 


60 531 


60 638 


60 746 


60 


853 


60 959 


61 066 


61 172 


61 278 
62 325 
63 347 


61 384 
62 428 
63 448 


61 490 
62 531 
63 548 


61 59.5 
62 634 
63 649 


61 700 
62 737 
63 749 


61 805 
62 839 
63 849 


61 
62 
63 


909 
941 
949 


62 014 
63 043 
64 048 


62 118 
63 144 
64 147 


62 221 
63 246 
64 246 


44 
45 
46 


64 345 
65 321 
66 276 


64 444 
65 418 
66 370 


64 542 
65 514 
66 464 


64 640 
65 610 
66 558 


64 738 
65 706 
66 652 


64 836 
65 801 
66 745 


64 
65 
66 


933 
896 
839 


65 031 
65 992 
66 932 


65 128 
66 087 
67 025 


65 225 
66 181 
67 117 


47 
48 
49 

59 


67 210 
68 124 
69 020 


67 302 
68 215 
69 108 


67 394 
68 305 
69 197 


67 486 
68 395 
69 285 


67 578 
68 485 
69 373 


67 669 
68 574 
69 461 


67 
68 
69 


761 
664 
548 


67 852 
68 753 
69 636 


67 943 
68 842 
69 723 


68 034 
68 931 
69 810 


69 897 


69 984 


70 070 


70 157 


70 243 


70 329 


70 


414 


70 501 


70 586 


70 672 


N. 


L. 


1 


2 


3 


4 


5 




6 


7 


8 


9 



16 



TABLE I 



50-100 



N. 


L. 




1 


2 


3 


4 


5 


6 


7 


8 


9 


50 


69 897 


69 


984 


70 070 


70 157 


70 243 


70 329 


70 415 


70 501 


70 586 


70 672 


51 


70 757 


70 


842 


70 927 


71 012 


71 096 


71 181 


71 265 


71 349 


71 433 


71 517 


52 


71 600 


71 


684 


71 767 


71 850 


71 933 


72 016 


72 099 


72 181 


72 263 


72 346 


53 


72 428 


72 


509 


72 591 


72 673 


72 754 


72 83o 


72 916 


72 997 


73 078 


73 159 


54 


73 239 


73 


320 


73 400 


73 480 


73 560 


73 640 


73 719 


73 799 


73 878 


73 957 


55 


74 036 


74 


115 


74 194 


74 273 


74 351 


74 429 


74 507 


74 5S6 


74 663 


74 741 


56 


74 819 


74 


896 


74 974 


75 051 


75 128 


75 205 


75 282 


75 358 


75 435 


75 511 


57 


75 587 


75 


664 


75 740 


75 815 


75 891 


75 967 


76 042 


76 118 


76 193 


76 268 


58 


76 343 


76 


418 


76 492 


76 567 


76 641 


76 716 


76 790 


76 864 


76 938 


77 012 


59 


77 085 


77 


159 


77 232 


77 305 


77 379 


77 452 


77 525 


77 597 


77 670 


77 743 


60 


77 815 


77 


887 


77 960 


78 032 


78 104 


78 176 


78 247 


78 319 


78 390 


78 462 


61 


78 533 


78 


604 


78 675 


78 746 


78 817 


78 888 


78 958 


79 029 


79 099 


79 169 


62 


79 239 


79 


309 


79 379 


79 449 


79 518 


79 588 


70 657 


79 727 


79 796 


79 865 


63 


79 934 


80 


003 


80 072 


80 140 


80 209 


80 277 


80 346 


80 414 


80 482 


80 550 


64 


80 618 


80 


686 


80 754 


80 821 


80 889 


80 956 


81 023 


81 090 


81 158 


81 224 


65 


81 291 


81 


358 


81 425 


81 491 


81 558 


81 624 


81 690 


81 757 


81 823 


81 889 


66 


81 954 


82 


020 


82 086 


82 151 


82 217 


82 282 


82 347 


82 413 


82 478 


82 543 


67 


82 607 


82 


672 


82 737 


82 802 


82 866 


82 930 


82 005 


83 050 


83 123 


83 187 


68 


83 251 


83 


315 


83 378 


83 442 


83 50(5 


83 569 


83 632 


83 696 


83 759 


83 822 


69 


83 885 


83 


948 


81 Oil 


84 073 


84 136 


84 198 


84 261 


84 323 


84 386 


84 448 


70 


84 510 


84 


572 


84 634 


84 696 


84 757 


84 819 


84 880 


84 942 


85 003 


85 065 


71 


85 126 


85 


187 


85 248 


85 309 


85 370 


85 431 


85 401 


85 552 


85 612 


85 673 


72 


85 733 


85 


794 


85 854 


85 914 


85 974 


S6 034 


86 004 


86 153 


86 213 


86 273 


73 


86 332 


86 


392 


86 451 


86 510 


86 570 


86 629 


86 688 


86 747 


86 806 


86 864 


74 


86 923 


86 


982 


87 040 


87 099 


87 157 


87 216 


87 274 


87 332 


87 300 


87 448 


75 


87 506 


87 


564 


87 622 


87 679 


87 737 


87 795 


87 852 


87 010 


87 067 


88 024 


76 


88 081 


88 


138 


88 195 


88 252 


88 309 


88 366 


88 423 


88 480 


88 536 


88 593 


77 


88 649 


88 


705 


88 762 


88 818 


88 874 


88 930 


88 086 


80 042 


89 098 


89 154 


78 


89 209 


89 


265 


89 321 


89 376 


89 432 


89 487 


89 542 


80 507 


89 653 


89 708 


79 


89 763 


89 


818 


89 873 


89 927 


89 982 


90 037 


00 001 


90 146 


90 200 


90 255 


80 


90 309 


90 


363 


90 417 


90 472 


90 526 


90 580 


00 634 


90 687 


00 741 


90 795 


81 


90 849 


90 


902 


90 956 


91 009 


91 062 


91 116 


01 160 


01 222 


01 275 


91 328 


82 


91 381 


91 


434 


91 487 


91 540 


91 593 


91 645 


91 608 


01 751 


01 803 


91 855 


83 


91 908 


91 


960 


92 012 


92 065 


92 117 


92 169 


02 221 


02 273 


02 324 


92 376 


84 


92 428 


92 


480 


92 531 


92 583 


92 634 


92 686 


02 737 


02 788 


92 840 


92 891 


85 


92 942 


92 


993 


93 044 


93 095 


93 146 


93 197 


93 247 


03 208 


93 349 


93 399 


86 


93 450 


93 


500 


93 551 


93 601 


93 651 


93 702 


03 752 


93 802 


03 852 


93 902 


87 


93 952 


94 


002 


94 052 


94 101 


94 151 


94 201 


04 250 


94 300 


04 349 


94 399 


88 


94 448 


94 


498 


94 547 


94 59C 


94 64, r > 


94 694 


94 743 


94 792 


94 841 


94 890 


89 


94 939 


94 


988 


95 036 


95 085 


95 134 


95 182 


95 231 


05 279 


95 328 


95 376 


90 


95 424 


95 


472 


95 521 


95 569 


95 617 


95 665 


95 713 


95 761 


95 809 


95 856 


91 


95 904 


95 


952 


95 999 


96 047 


96 095 


06 142 


96 190 


96 237 


96 284 


96 332 


92 


96 379 


96 




96 473 


96 520 


96 567 


96 614 


06 661 


96 708 


96 755 


96 802 


93 


96 848 


96 


895 


96 942 


96 988 


97 035 


97 081 


07 128 


97 174 


07 220 


97 267 


94 


97 313 


97 


359 


97 405 


97 451 


97 407 


97 543 


07 589 


97 635 


97 681 


97 727 


95 


97 772 


97 


818 


97 864 


97 909 


97 955 


08 000 


98 046 


98 091 


98 137 


98 182 


96 


98 227 


98 


272 


98 318 


98 363 


98 408 


98 453 


98 498 


98 543 


98 588 


98 632 


97 


98 677 


98 


722 


98 767 


98 811 


98 856 


08 900 


98 945 


98 989 


99 034 


99 078 


98 


99 123 


99 


167 


99 211 


99 255 


99 300 


99 344 


90 388 


99 432 


99 476 


99 520 


99 


99 564 


99 


607 


99 651 


99 695 


99 739 


99 782 


99 826 


99 870 


99 913 


99 957 


100 


00 000 


00 


043 


00 087 


00 130 


00 173 


00 217 


00 260 


00 303 


00 346 


00 389 


N. 


L. 




1 


2 


3 


4 


5 


6 


7 


8 


9 



17 



TABLE I 



100-150 



N. 


L. o | 


i 


3 


3 1 


4 














Prop. Parti 


100 


00 000 


043 


087 


130 


173 


217 


260 


303 


346 


389 






101 


432 


475 


518 


561 


604 


647 


689 


732 


775 


817 




44 43 42 


102 


860 


903 


945 


988 


*030 


*072 


"115 


"157 


"199 


*242 


1 


4.4 4.3 4.2 


.103 


01 284 


326 


368 


410 


452 


494 


536 


578 


620 


662 


2 


8.8 8.6 8.4 


104 


703 


745 


787 


828 


870 


912 


953 


995 


*036 


*078 


3 


13.2 12.9 12.6 


105 


02 119 


160 


202 


243 


284 


325 


366 


407 


449 


490 


4 


17.6 17.2 16.8 

It f\ tl C 11 f\ 


106 
107 


531 
938 


572 
979 


612 
*019 


653 
*060 


694 
*100 


735 
*141 


776 
*181 


816 
*222 


857 
*262 


898 
*302 


6 


22.0 21.5 21.0 
26.4 25.8 25.2 


108 
109 


03 342 
743 


383 
782 


423 
822 


463 
862 


503 
902 


543 
941 


583 
981 


623 
*021 


663 
*060 


703 
*100 


8 
9 


30.8 30.1 29.4 
35.2 34.4 33.6 
39.6 38.7 37.8 


110 


04 139 


179 


218 


258 


297 


336 


376 


415 


454 


493 






111 


532 


571 


610 


650 


689 


727 


766 


805 


844 


883 




41 40 39 


112 


922 


961 


999 


*038 


*077 


*115 


*154 


*192 


*231 


*269 


1 


4.1 4.0 3.9 


113 


05 308 


346 


385 


423 


461 


500 


538 


576 


614 


652 


2 


8.2 8.0 7.8 


114 


690 


729 


767 


805 


843 


881 


918 


956 


994 


*032 


3 


12.3 12.0 11.7 


115 

116 
117 
118 
119 


06 070 
446 
819 
07 188 
555 


108 
483 
856 
225 
591 


145 
521 
893 
262 
628 


183 
558 
930 
298 
664 


221 
595 
967 
335 
700 


258 
633 
*004 
372 
737 


296 
670 
*041 
408 
773 


333 
707 
*078 
445 
809 


371 
744 
*1I5 
482 
846 


408 
781 
*I51 
518 
882 


4 
5 
6 
7 
8 
9 


16.4 16.0 15.6 
20.5 20.0 19.5 
24.6 24.0 23.4 
28.7 28.0 27.3 
32.8 32.0 31.2 
36.9 36.0 35. 1 


120 


918 


954 


990 


*027 


*063 


*099 


*135 


*171 


*207 


*243 






121 


08 279 


314 


350 


386 


422 


458 


493 


529 


565 


600 




38 37 36 


122 


636 


672 


707 


743 


778 


814 


849 


884 


920 


955 


1 


3.8 3.7 3.6 


123 


991 


*026 


*061 


*096 


*132 


*167 


*202 


*237 


*272 


*307 


2 


7.6 7.4 7.2 


124 


09 342 


377 


412 


447 


482 


517 


552 


587 


621 


656 


3 


11.4 11.1 10.8 


125 


691 


726 


760 


795 


830 


864 


899 


934 


968 


*003 


4 


15.2 14.8 14.4 


126 
127 
128 
129 


10 037 
380 
721 
11 059 


072 
415 
755 
093 


106 
449 
789 
126 


140 
483 
823 
160 


175 
517 
857 
193 


209 
551 
890 
227 


243 
585 
924 
261 


278 
619 
958 
294 


312 
653 
992 
327 


346 
687 
*025 
361 


6 
7 
8 
9 


19.0 18.5 18.0 
22.8 22.2 21.6 
26.6 25.9 25 2 
30.4 29.6 28 8 
34.2 33.3 32.4 


130 


394 


428 


461 


494 


528 


561 


594 


628 


661 


694 






131 


727 


760 


793 


826 


860 


893 


926 


959 


992 


*024 




35 34 33 


132 


12 057 


090 


123 


156 


189 


222 


254 


287 


320 


352 


1 


3.5 3.4 3.3 


133 


385 


418 


450 


483 


516 


548 


581 


613 


646 


678 


2 


7.0 6.8 6.6 


134 


710 


743 


775 


808 


840 


872 


905 


937 


969 


*001 


3 


10.5 10.2 9.9 


135 

136 
137 


13 033 
354 
672 


066 
386 
704 


098 
418 
735 


130 
450 
767 


162 
481 
799 


194 
513 
830 


226 
545 
862 


258 
577 
893 


290 
609 
925 


322 
640 
956 


4 
5 
6 


14.0 13.6 13.2 
17.5 17.0 16 5 
21.0 20.4 19.8 

f\ A e |7 o J3 | 


138 


988 


*019 


*051 


*082 


*114 


*145 


*176 


*208 


*239 


*270 




24.5 23.o 23. 1 


139 


14 301 


333 


364 


395 


426 


457 


489 


520 


551 


582 


9 


28.0 27.2 26.4 
31.5 30.6 29.7 


140 


613 


644 


675 


706 


737 


768 


799 


829 


860 


891 






141 


922 


953 


983 


*014 


*045 


*076 


*106 


*137 


*168 


*198 




32 31 30 


142 


15 229 


259 


290 


320 


351 


381 


412 


442 


473 


503 




3.2 3.1 3.0 


143 


534 


564 


594 


625 


655 


685 


715 


746 


776 


806 


2 


6.4 6.2 6.0 


144 


836 


866 


897 


927 


957 


987 


*017 


*047 


*077 


*107 


3 


9.6 9.3 9.0 


146 
146 
147 
148 
149 


16 137 
435 
732 
17 026 
319 


167 
465 
76 
056 
348 


197 
495 
791 
085 
377 


227 
524 
820 
114 
406 


256 
554 
850 
143 
435 


286 
584 
879 
173 
464 


316 
613 
909 
202 
493 


546 
643 
938 
231 
522 


376 
673 
967 
260 
551 


406 
702 
997 
289 
580 


4 
5 
6 
7 
8 
9 


12.8 12.4 12.0 
16.0 15.5 15 
19.2 18.6 18.0 
22.4 21.7 21 
25.6 24.8 24 
28.8 27.9 27.0 


150 


609 


638 


667 


696 


725 


754 


782 


811 


840 


869 






If. 


L. o 


z 


a 


3 


4 


5 


6 


7 


8 


1 * 




Prop. Parts 



IS 



TABLE I 



150-200 



N. 




I | 2 3 




6 | 7 8 | 9 | Prop. Parts 


150 


7 609 


638 


667 


696 


725 


754 


782 


811 


840 


869 




151 


898 


926 


955 


984 


*013 


041 


*070 


*099 


*127 


*156 


29 28 


152 


8 184 


213 


241 


270 


298 


327 


355 


384 


412 


441 


1 


z.v z.o 


153 


469 


498 


526 


554 


583 


611 


639 


667 


696 


724 


2 


5.8 5.6 


154 


752 


780 


808 


837 


865 


893 


921 


949 


977 


*005 


3 


8.7 8.4 


165 


9 033 


061 


089 


117 


145 


173 


201 


229 


257 


283 


4 


11.6 11.2 
M.5 14.0 


156 


312 


340 


368 


396 


424 


451 


479 


507 


535 


562 




J7 4 16 8 


157 


590 


618 


645 


673 


700 


728 


756 


783 


811 


838 


7 


20 3 19.6 


158 


866 


893 


921 


948 


976 


*003 


*030 


*058 


*083 


*112 


8 


23 2 22 4 


159 


20 140 


167 


194 


222 


249 


276 


303 


330 


358 


383 


9 


26! 1 25!2 


160 


412 


439 


466 


493 


520 


548 


575 


602 


629 


656 




Af* AA 


161 


683 


710 


737 


763 


790 


817 


844 


871 


898 


925 


Zf 20 


162 


952 978 


*005 


*032 


*059 


*085 


*112 


"139 


*165 


*192 


1 


Z./ <.o 

54 C 1 


163 


21 219 


245 


272 


299 


325 


352 


37* 


405 


431 


458 


2 


.4 5.2 


164 


484 


511 


537 


564 


590 


617 


643 


669 


696 722 


3 


8.1 7.8 
i A o in A 


165 


748 


775 


801 


827 


854 


880 


906 


932 


958 


985 


4 
5 


lu.o 10.4 
13.5 13.0 


166 


22 Oil 


037 


063 


089 


115 


141 


167 


194 


220 


246 




16 2 15.6 


167 


272 


298 


324 


350 


376 


401 


427 


453 


479 


505 


7 


18.9 18.2 


168 


531 


5^7 


583 


608 


634 


660 


686 


712 


737 


763 




21 6 20 8 


169 


789 


814 


840 


866 


891 


917 


943 


968 


994 


*019 


9 


24 '.3 23^4 


170 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 




AF 


171 


300 


325 


350 


376 


401 


426 


452 


477 


502 


528 


3D 

I/I c 


172 


553 


578 


603 


629 


654 


679 


704 


729 


754 


779 


2.5 

2C f\ 


173 


805 


830 


855 


880 


905 


930 


955 


980 


*005 


*030 


5.0 

37 *i 


174 


24 053 


080 


103 


130 


155 


180 


204 


229 


254 


279 


/.-> 

A 1ft ft 


175 


304 


329 


353 


378 


403 


428 


452 


477 


502 


527 


1 1 U U 

5 12.5 


176 


551 


576 


601 


625 


650 


674 


699 


724 


748 


773 


6 15 


177 


797 


822 


846 


871 


895 


920 


944 


969 


993 


*018 


7 17 5 


178 


25 042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


8 20 


179 


285 


310 


334 


358 


382 


406 


431 


455 


479 


503 


9 22.5 


180 


527 


551 


575 


600 


624 


648 


672 


696 


720 


744 


ni 00 


181 


768 


792 


81 


84 


864 


888 


912 


935 


959 


983 




At *n 

2 A 1 ^ 


182 


26 007 


031 


05 


07 


102 


12 


150 


174 


198! 221 




.T L'J 
40 A 


183 


245 


269 


29 


31 


34 


36 


387 


41 


435! 458 


2 


.O 4.0 
71 A Q 


184 


482 


505 


52 


55 


57 


60 


623 


64 


670 


694 


3 


.L 0.7 
9 A 7 


185 


717 


741 


76 


78 


81 


83 


858 


88 


903 


92 


5 


.0 V. L 

120 11.5 


186 


95 


975 


99 


*02 


*04 


*06 


*09 


*11' 


*138 


*16 


6 


14.4 13.8 


187 


27 184 


207 


23 


25 


27 


30 


323 


34 


370 


39 


7 


16 8 16.1 


188 


416 


439 


46 


48 


50 


53 


554 


57 


600 


62 


8 


19 2 18 4 


189 


64 


669 


69 


71 


73 


76 


784 


80 


830 


85 


9 


2\'.6 20.7 


190 


87 


89' 


92 


94 


96 


98 


*01 


*03 


*058 


*08 




99 21. 


191 


28 10 


12* 


14 


17 


19 


21 


24C 


26 


283 


30 




21 1 I 


192 


33 


353 


37 


39 


42 


44 


46 


48 


511 


53 


1 


.2 L. I 

A A 47 


193 
194 


55 
78 


57* 
803 


60 
82 


62 
84 


64 
87 


66 
89 


69 
91 


71 
93 


735 
95$ 


75 
98 


2 

3 


6*.6 6'3 
ft ft ft 4 


195 


29 00 


02< 


04 


07 


09 


11 


13 


15 


181 


20 


5 


O . O o ~ 

11.0 10.5 


196 


22 


24 


27 


29 


31 


33 


35 


38 


40: 


\ 42 


6 


13 2 12.6 


197 


44 


46 


49 


51 


53 


55 


57 


60 


62: 


J 64 


7 


15.4 14.7 


198 


66 


68 


71 


73 


75 


77 


79 


82 


84, 


Z 86 


8 


17 6 16.8 


199 


88 


90 


92 


95 


97 


99 


*01 


*03 


*06( 


) *08 


9 


19'.8 18.9 


200 


30 10 


12 


14 


16 


19 


21 


23 


25 


27 


S 29 






N. 


L. o 




a 


3 


4 


5 


6 


7 


8 


9 


Prop, Parts 



10 



TABLE I 



200-250 



N. 


L. o | i a 


3 


4 


5 


6 


7 


8 


9 


Prop. Parts 


200 


30 1031 125 


146 


168 


190 


211 


233 


255 


276 


298 




201 


320 


341 


363 


384 


406 


428 


449 


471 


492 


514 


22 21 


202 


535 


557 


578 


600 


621 


643 


664 


685 


707 


728 


1 2.2 2.1 


203 


750 


771 


792 


814 


835 


856 


878 


899 


920 


942 


2 4.4 4.2 


204 


963 


984 


*006 


*027 


*048 


*069 


*091 


*I12 


*133 


*154 


3 6.6 6.3 


205 


31 175 


197 


218 


239 


260 


281 


302 


323 


345 


366 


4 8.8 8.4 

51 1 A in e 


206 


387 


408 


429 


450 


471 


492 


513 


534 


555 


576 


1 1 . V III. J 
61 7 1 11 A 


207 


597 


618 


639 


660 


681 


702 


723 


744 


765 


785 


Ij.Z IZ.O 
7 IS A 147 


208 


806 


827 


848 


869 


890 


911 


931 


952 


973 


994 


1 j . *f l*r. / 
817 A 1 A ft 


209 


32 015 


035 


056 


077 


098 


118 


139 


160 


181 


201 


I/.O 10. 

9 19.8 18.9 


210 


222 


243 


263 


284 


305 


325 


346 


366 


387 


408 




211 


428 


449 


469 


490 


510 


531 


552 


572 


593 


613 


20 


212 


634 


654 


675 


695 


715 


736 


756 


777 


797 


818 


1 


2.0 


213 


838 


858 


879 


899 


919 


940 


960 


980 


*001 


*021 


2 


4.0 


214 


33 041 


062 


082 


102 


122 


143 


163 


183 


203 


224 


3 


6.0 


215 


244 


264 


284 


304 


325 


345 


365 


385 


405 


425 


4 


8.0 

i A A 


216 


445 


465 


486 


506 


526 


546 


566 


586 


606 


626 




10. 

1 1 A 


217 
218 


646 
846 


666 
866 


686 
885 


706 
905 


726 
925 


746 
945 


766 
965 


786 
985 


806 
*005 


826 
*025 


7 


12.0 
14.0 

\f A 


219 


34 044 


064 


084 


104 


124 


143 


163 


183 


203 


223 


9 


lo. U 
18.0 


220 


242 


262 


282 


301 


321 


341 


361 


380 


400 


420 






221 


439 


459 


479 


498 


518 


537 


557 


577 


596 


616 


19 


222 


635 


655 


674 


694 


713 


733 


753 


772 


792 


11 


1 


1.9 


223 


830 


850 


869 


889 


908 


928 


947 


967 


986 


*005 


2 


3.8 


224 


35 025 


044 


064 


083 


102 


122 


141 


160 


180 


199 


3 


5.7 


225 


218 


238 


257 


276 


295 


315 


334 


353 


372 


392 


4 


7.6 

9c 


226 


411 


430 


449 


468 


488 


507 


526 


545 


564 


583 




.5 


227 


603 


622 


641 


660 


679 


698 


717 


736 


755 


774 




1 1.4 
17 7 


228 


793 


813 


832 


851 


870 


889 


908 


927 


946 


965 




13.3 


229 


984 


*003 


*021 


*040 


*059 


*078 


*097 


*116 


*135 


*154 


8 
9 


15.2 
17.1 


230 


36 173 


192 


211 


229 


248 


267 


286 


305 


324 


342 






231 


361 


380 


399 


418 


436 


455 


474 


493 


511 


530 


18 


232 


549 


568 


586 


605 


624 


642 


661 


680 


698 


717 


1 


1.8 


233 


736 


754 


773 


791 


810 


829 


847 


866 


884 


903 


2 


3.6 


234 


922 


940 


959 


977 


996 


*014 


*033 


*051 


*070 


*088 


3 


5.4 


235 


37 107 


125 


144 


162 


181 


199 


218 


236 


254 


273 


4 


7.2 
9/t 


236 


291 


310 


328 


346 


365 


383 


401 


420 


438 


457 




.0 

1 A O 


237 


475 


493 


511 


530 


548 


566 


585 


603 


621 


639 




10 o 

t t A 


238 


658 


676 


694 


712 


731 


749 


767 


785 


803 


822 




IZ. 

Mt 


239 


840 


858 


876 


894 


912 


931 


949 


967 


985 


*003 


9 


.4 
16.2 


240 


38 021 


039 


057 


075 


093 


112 


130 


148 


166 


184 






241 


202 


220 


238 


256 


274 


292 


310 


328 


346 


364 


17 


242 


382 


399 


417 


435 


453 


471 


489 


507 


525 


543 


1 


1.7 


243 


561 


578 


596 


614 


632 


650 


668 


686 


703 


721 


2 


3.4 


244 


739 


757 


775 


792 


810 


828 


846 


863 


881 


899 


3 


5.1 


246 


917 


934 


952 


970 


987 


*005 


*023 


*041 


*058 


*076 


4 


6.8 

8e 


246 


39 094 


111 


129 


146 


164 


182 


199 


217 


235 


252 




.3 

1 A -J 


247 


270 


287 


305 


322 


340 


358 


375 


393 


410 


428 




lu.z 
no 


248 


445 


463 


480 


498 


515 


533 


550 


568 


585 


602 




.7 
1 1 A 


249 


620 


637 


655 


672 


690 


707 


724 


742 


759 


777 


9 


13.0 
15.3 


250 


794 


811 


829 


846 


863 


881 


898 


915 


933 


950 






N. 


L. o 


i 


a 


3 4 


5 


6 


7 


8 


9 Prop. Parts 



20 



TABLE I 



250-300 



N. 


















Drsk Darta 


260 


39 794 


811 


829 


846 


863 


5 

881 


898 


7 

915 


933 


9 

950 


jfrop. jrara 
18 


251 


967 


985 


*002 


*019 


*037 


*054 


*071 


*088 


*106 


*123 


1 


1 8 


252 


40 140 


157 


175 


192 


209 


226 


243 


261 


278 


295 


2 


3.6 


253 


312 


329 


346 


364 


381 


398 


415 


432 


449 


466 


3 


5 4 


254 


483 


500 


518 


535 


552 


569 


586 


603 


620 


637 


4 


7.2 


255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


5 


9.0 


256 


824 


841 


858 


875 


892 


909 


926 


943 


960 


976 


6 


10.8 


257 


993 


*010 


*027 


*044 


*061 


*078 


*095 


*111 


*128 


*145 


7 


12.6 


258 


41 162 


179 


196 


212 


229 


246 


263 


280 


296 


313 


8 


14.4 


259 


330 


347 


363 


380 


397 


414 


430 


447 


464 


481 


9 


16.2 


260 


497 


514 


531 


547 


564 


581 


597 


614 


631 


647 


17 


261 


664 


681 


697 


714 


731 


747 


764 


780 


797 


814 


1 


1 7 


262 


830 


847i 863 


880 


896 


913 


929 


946 


963 


979 


2 


3 4 


263 


996 


*QU 


*029 


*045 


*062 


*078 


*095 


*111 


*127 


*144 


3 


5 1 


264 


42 160 


177 


193 


210 


226 


243 


239 


275 


292 


308 


4 


6.8 


265 


323 


341 


357 


374 


390 


406 


423 


439 


455 


472 


5 


8.5 


266 


488 


504 


521 


537 


553 


570 


586 


602 


619 


635 


6 


10.2 


267 


651 


667 


684 


700 


716 


732 


749 


765 


781 


797 


7 


11.9 


268 


813 


8*J 


846 


862 


878 


894 


911 


927 


943 


959 


8 


13.6 


269 


975 


991 


*008 


*024 


*040 


*056 


*072 


*088 


*104 


*120 


9 


15.3 


270 


43 136 


152 


169 


185 


201 


217 


233 


249 


265 


281 


loge = 0.43429 


271 


297 


313 


329 


345 


361 


377 


393 


409 


425 


441 


16 


272 


457 


473 


489 


505 


521 


537 


553 


569 


584 


600 


1 


1 6 


273 


616 


632 


648 


664 


680 


696 


712 


727 


743 


759 


2 


3.2 


274 


775 


791 


807 


823 


838 


854 


870 


886 


902 


917 


3 


4.8 


276 


933 


949 


965 


981 


996 


*012 


*028 


*044 


*059 


*075 


4 


6.4 


276 


44 091 


107 


122 


138 


154 


170 


185 


201 


217 


232 


5 


8.0 


277 


248 


264 


279 


295 


311 


326 


342 


358 


373 


389 


6 


9.6 


278 


404 


420 


436 


451 


467 


483 


498 


514 


529 


545 


7 


11.2 


279 


560 


576 


592 


607 


623 


638 


654 


669 


685 


700 


8 


12 8 


280 


716 


731 


747 


762 


778 


793 


809 


824 


840 


855 


9 


14.4 


281 


871 


886 


902 


917 


932 


948 


963 


979 


994 


*010 


15 


282 


45 025 


040 


056 


071 


086 


102 


117 


133 


148 


163 


1 


1 5 


283 


179 


194 


209 


225 


240 


255 


271 


286 


301 


317 


2 


3.0 


284 


332 


347 


362 


378 


393 


408 


423 


439 


454 


469 


3 


4.5 


285 


484 


500 


515 


530 


545 


561 


576 


591 


606 


621 


4 


6.0 


286 


637 


652 


667 


682 


697 


712 


728 


743 


758 


773 


5 


7.5 


287 


788 


803 


818 


834 


849 


864 


879 


894 


909 


924 


6 


9.0 


288 


939 


954 


969 


984 


*000 


*015 


*030 


*045 


*060 


*075 


7 


10.5 


289 


46 090 


105 


120 


135 


150 


165 


180 


195 


210 


225 


8 


12.0 


290 


240 


255 


270 


285 


300 


315 


330 


345 


359 


374 


9 


13.5 


291 


389 


404 


419 


434 


449 


464 


479 


494 


509 


523 


14 


292 


538 


553 


568 


583 


598 


613 


627 


642 


657 


672 


1 


1.4 


293 


687 


702 


716 


731 


746 


761 


776 


790 


805 


820 


2 


2.8 


294 


835 


850 


864 


879 


894 


909 


923 


938 


953 


967 


3 


4.2 


295 


982 


997 


*012 


*026 


*041 


*056 


*070 


*085 


*100 


*114 


4 


5.6 


296 


47 129 


144 


159 


173 


188 


202 


217 


232 


246 


261 


5 


7.0 


297 


276 


290 


305 


319 


334 


349 


363 


378 


392 


407 


6 


8.4 


298 


422 


436 


451 


465 


480 


494 


509 


524 


538 


553 


7 


9.8 


299 


567 


582 


596 


611 


625 


640 


654 


669 


683 


698 


8 


11.2 


300 


712 


727 


741 


756 


770 


784 


799 


813 


828 


842 


9 


12.6 


N. 


L. o 


X 


a 


3 


4 


s 


6 


7 I 


9 


Prop. Parts 



TABLE I 



300-350 



IT. 


L. 


z 


a 


3 


4 


5 


6 


7 


8 


9 


Prop. Parts 


800 


47 712 


727 


741 


756 


770 


784 


799 


813 


828 


842 




301 


857 


871 


885 


900 


914 


929 


943 


958 


972 


986 




302 


48 001 


015 


029 


044 


058 


073 


087 


101 


116 


130 


4 m 


303 


144 


159 


173 


187 


202 


216 


230 


244 


259 


273 




JLO 

If 


304 


287 


302 


316 


330 


344 


359 


373 


387 


401 


416 


I 


.5 

1 A 


805 


430 


444 


458 


473 


487 


501 


515 


530 


544 


558 


^ 


J U 

4 5 


306 


572 


586 


601 


615 


629 


643 


657 


671 


686 


700 


4 


6.0 


307 


714 


728 


742 


756 


770 


785 


799 


813 


827 


841 




7 ^ 


308 


855 


869 


883 


897 


911 


926 


940 


954 


968 


982 


? 


/ . j 

Q n 


309 


996 


*010 


*024 


*038 


*052 


*066 


*080 


*094 


*108 


*122 


7 


r . U 

10.5 


310 


49 136 


150 


164 


178 


192 


206 


220 


234 


248 


262 


8 


12.0 


311 


276 


290 


304 


318 


332 


346 


360 


374 


388 


402 


9 


13.5 


312 


415 


429 


443 


457 


471 


485 


499 


513 


527 


541 






313 


554 


568 


582 


596 


610 


624 


638 


651 


665 


679 


1<nr ir n 4Q715 


314 


693 


707 


721 


734 


748 


762 


776 


790 


803 


817 


l\j 71 *\J.l7i U 


315 


831 


843 


859 


872 


886 


900 


914 


927 


941 


955 




316 


969 


982 


996 


*010 


*024 


*037 


*051 


*065 


*079 


*092 


1 I A 


317 


50 106 


120 


133 


147 


161 


174 


188 


202 


215 


229 


\ 


i i 

20 


318 


243 


256 


270 


?84 


297 


311 


325 


338 


352 


365 


l 


. 

4 ? 


319 


379 


393 


4C6 


420 


433 


447 


461 


474 


488 


501 


j 

4 


f . L 

5 6 


320 


515 


529 


542 


556 


569 


583 


596 


610 


623 


637 


5 


7 


321 


651 


664 


678 


691 


705 


718 


732 


745 


759 


772 


6 


8 4 


322 


786 


799 


813 


826 


840 


853 


866 


880 


893 


907 


7 


9 8 


323 


920 


934 


947 


961 


974 


937 


*001 


*014 


*028 


*041 


9 


11 2 


324 


51 055 


068 


081 


095 


108 


121 


135 


148 


162 


175 


9 


12.6 


325 


188 


202 


215 


228 


242 


255 


268 


282 


295 


308 




326 


322 


335 


348 


362 


375 


388 


402 


415 


428 


441 




327 


455 


468 


481 


495 


508 


521 


534 


548 


561 


574 


13 


328 


587 


601 


614 


627 


640 


654 


667 


680 


693 


706 




1 3 


329 


720 


733 


746 


759 


772 


786 


799 


812 


825 


838 


2 


2 6 


330 


851 


865 


878 


891 


904 


917 


930 


943 


957 


970 


3 


3 9 


331 


983 


996 


*009 


*022 


*035 


*048 


*061 


*075 


*088 


*I01 


4 


5 2 


332 


52 114 


127 


140 


153 


166 


179 


192 


205 


218 


231 


5 


6 5 


333 


244 


257 


270 


284 


297 


310 


323 


336 


349 


362 


6 


7.8 


334 


375 


388 


401 


414 


427 


440 


453 


466 


479 


492 


7 


9.1 


336 


504 


517 


530 


543 


556 


569 


582 


595 


608 


621 


8 


10.4 


336 


634 


647 


660 


673 


686 


699 


711 


724 


737 


750 


9 


11.7 


337 


763 


776 


789 


802 


813 


827 


840 


853 


866 


879 




338 


892 


905 


917 


930 


943 


956 


969 


982 


994 


*007 




339 


53 020 


033 


046 


058 


071 


084 


097 


110 


122 


135 


12 


340 


148 


161 


173 


186 


199 


212 


224 


237 


250 


263 


1 


1.2 


341 


275 


288 


301 


314 


326 


339 


352 


364 


377 


390 


2 


2.4 


342 


403 


415 


428 


441 


453 


466 


479 


491 


504 


517 


3 


3 6 


343 


529 


542 


555 


567 


580 


593 


605 


618 


631 


643 


4 


4 8 


344 


656 


668 


681 


694 


706 


719 


732 


744 


757 


769 


5 


6.0 


345 


782 


794 


807 


820 


832 


845 


857 


870 


882 


895 


6 


7.2 

8 A 


346 


908 


920 


933 


945 


958 


970 


983 


995 


*008 


*020 




4 
9f. 


347 


54 033 


045 


058 


070 


083 


095 


108 


120 


133 


145 




.0 
10 R 


348 


158 


170 


183 


195 


208 


220 


233 


245 


258 


270 




1 U. O 


349 


283 


295 


307 


320 


332 


345 


357 


370 


382 


394 




360 


407 


419 


432 


444 


456 


469 


481 


494 


506 


518 




N. 


L. o 


i 


2 


3 


4 


5 


6 | 7 


8 9 


Prop. Parts 



22 



TABLE I 



350-400 



N. L. 


i 


3345 


6 | f 


8 


9 


Prop. Parts 


850 


54 407 


419 


432 


444 


456 


469 


481 


494 


506 


518 




351 


531 


543 


555 


568 


580 


593 


605 


617 


630 


642 




352 


654 


667 


679 


691 


704 


716 


728 


741 


753 


765 




353 


777 


790 


802 


814 


827 


839 


851 


864 


876 


888 


4 


354 


900 


913 


925 


937 


949 


962 


974 


986 


998 


Oil 


1 


49 

1.3 


366 


55 023 


035 


047 


060 


072 


084 


096 


108 


121 


133 


2 


2 6 


356 


145 


157 


169 


182 


194 


206 


218 


230 


242 


255 


3 


3.9 


357 


267 


279 


291 


303 


315 


328 


340 


352 


364 


376 


4 


5 2 


358 


388 


400 


413 


425 


437 


449 


461 


473 


485 


497 


5 


6.5 


359 


50<? 522 


534 


546 


558 


570 


582 


594 


606 


618 





7.8 


360 


6, fA2 


654 


666 


678 


691 


703 


71") 


727 


739 


7 


9.1 


361 


751 763 


775 


787 


799 


811 


823 


s>5 


847 


859 


8 


10.4 


362 


871 88* 


895 


907 


919 


931 


943 


oS 


967 


979 


9 11.7 


363 


991 


^003 


*015 


*027 


*038 


*050 


*062 


074 


*086 


*098 




364 


56 110 


122 


134 


146 


158 


170 


182 


194 


205 


217 




366 


229 


241 


253 


265 


277 


289 


301 


312 


324 


336 


12 


366 


348 


360 


372 


384 


396 


407 


419 


431 


443 


455 


1 


1.2 


367 


467 


478 


490 


502 


514 


526 


538 


549 


561 


573 


2 


2.4 


368 


585 


507 


608 


620 


632 


644 


656 


667 


679 


691 


3 


3 6 


369 


703 


y,4 


726 


738 


750 


761 


773 


785 


797 


808 


4 


4.8 


370 


820 


832 


844 


855 


867 


879 


891 


902 


914 


926 


5 


6 


371 


937 


949 


961 


972 


984 


996 


*008 


*019 


*031 


*043 


6 


7.2 


372 


57 054 


066 


078 


089 


101 


113 


124 


136 


148 


159 


7 


8.4 


373 


171 


183 


194 


206 


217 


229 


241 


252 


264 


276 


8 


9.6 


374 


287 


299 


310 


322 


334 


345 


357 


368 


380 


392 


9 


10.8 


375 


403 


415 


426 


438 


449 


461 


473 


484 


496 


507 




376 


519 


530 


542 


553 


565 


576 


588 


600 


611 


623 




377 


634 


646 


657 


669 


680 


692 


703 


715 


726 


738 


11 


378 


749 


761 


772 


784 


795 


807 


818 


830 


841 


852 


1 


1. 1 


379 


864 


875 


887 


898 


910 


921 


933 


944 


955 


967 


2 
7 


2.2 
\ \ 


380 


978 


990 


*001 


*013 


*024 


*035 


*047 


*058 


*070 


*081 


j 
4 


j . j 
4.4 


381 


58 092 


104 


115 


127 


138 


149 


161 


172 


184 


195 


5 


5.5 


382 


- 206 


218 


229 


240 


252 


263 


274 


286 


297 


309 


5 


6~6 


383 


320 


331 


343 


354 


365 


377 


388 


399 


410 


422 


7 


77 


384 


433 


444 


456 


467 


478 


490 


501 


512 


524 


535 


8 


8 8 


386 


546 


557 


569 


580 


591 


602 


614 


625 


636 


647 


9 


9.9 


386 


659 


670 


681 


692 


704 


715 


726 


737 


749 


760 




387 


771 


782 


794 


805 


816 


827 


838 


850 


861 


872 




388 


883 


894 


906 


917 


928 


939 


950 


961 


973 


984 


in 


389 


995 


*006 


*017 


*028 


*040 


*051 


*062 


*073 


*084 


*095 


1 


1.0 


390 


59 106 


118 


129 


140 


151 


162 


173 


184 


195 


207 


2 


2.0 


391 


218 


229 


240 


251 


262 


273 


284 


295 


306 


318 


3 


3.0 


392 


329 


340 


351 


362 


373 


384 


395 


406 


417 


428 


4 


4.0 


393 


439 


450 


461 


472 


483 


494 


506 


517 


528 


539 


5 


5.0 


394 


550 


561 


572 


583 


594 


605 


616 


627 


638 


649 


6 


6.0 


396 


660 


671 


682 


693 


704 


715 


726 


737 


748 


759 


7 


7.0 


396 


770 


780 


791 


802 


813 


824 


835 


846 


857 


868 


8 


8.0 


397 


879 


890 


901 


912 


923 


934 


945 


956 


966 


977 


9 


9.0 


398 


988 


999 


*010 


*021 


*032 


*043 


*054 


*065 


*076 


*086 




399 


60 097 


108 


119 


130 


141 


152 


163 


173 


184 


195 




400 


206 


217 


228 


239 


249 


260 


271 


282 


293 


304 




W. L. o 


z 


2 


3 


4 


5 


6 


7 


8 


9 | Prop. Parts 



TABLE I 



400-450 



N. 


L. o | x | a 3 | 4 


5 


6 7 


8 | 9 Prop. Parts 


400 


60 206 


217 


228 


239 


249 


260 


271 


282 


293 


304 




401 


314 


325 


336 


347 


358 


369 


379 


390 


401 


412 




402 


423 


433 


444 


455 


466 


477 


487 


498 


509 


520 




403 


531 


541 


552 


563 


574 


584 


595 


606 


617 


627 




404 


638 


649 


660 


670 


681 


692 


703 


713 


724 


735 




406 


746 


756 


767 


778 


788 


799 


810 


821 


831 


842 


44 


406 


853 


863 


874 


885 


895 


906 


917 


927 


938 


949 




J.J. 

11 


407 


959 


970 


981 


991 


*002 


*013 


*023 


*034 


*045 


*055 




. 1 
29 


408 


61 066 


077 


087 


098 


109 


119 


130 


140 


151 


162 


^ 


. L 
\ \ 


409 


172 


183 


194 


204 


215 


225 


236 


247 


257 


268 


J 

4 


j . j 

4.4 


410 


278 


289 


300 


310 


321 


331 


342 


352 


363 


374 


5 


5.5 


411 


384 


395 


405 


416 


426 


437 


448 


458 


469 


479 


6 


66 


412 


490 


500 


511 


521 


532 


542 


553 


563 


574 


584 


7 


7 7 


413 


595 


606 


616 


627 


637 


648 


658 


669 


679 


690 


8 


8 8 


414 


700 


711 


721 


731 


742 


752 


763 


773 


784 


794 


9 


9.9 


415 


805 


815 


826 


836 


847 


857 


868 


878 


888 


899 




416 


909 


920 


930 


941 


951 


962 


972 


982 


993 


*003 




417 


62 014 


024 


034 


045 


055 


066 


076 


086 


097 


107 




418 


118 


128 


138 


149 


159 


170 


180 


190 


201 


211 




419 


221 


232 


242 


252 


263 


273 


284 


294 


304 


315 




420 


323 


335 


346 


356 


366 


377 


387 


397 


408 


418 




421 


428 


439 


449 


459 


469 


480 


490 


5CO 


511 


521 


10 


422 


531 


542 


552 


562 


572 


583 


593 


603 


613 


624 


1 


1.0 


423 


634 


644 


655 


665 


675 


685 


696 


706 


716 


726 


2 


2.0 


424 


737 


747 


757 


767 


778 


788 


798 


808 


818 


829 


3 


3.0 


425 


839 


849 


859 


870 


880 


890 


900 


910 


921 


931 


4 


4.0 

5 A 


426 


941 


951 


961 


972 


982 


992 


*002 


*012 


*022 


*033 




u 
6n 


427 


63 043 


053 


063 


073 


083 


094 


104 


114 


124 


134 




u 

7n 


428 


144 


155 


165 


175 


185 


195 


205 


215 


225 


236 




.u 

8 A 


429 


246 


256 


266 


276 


286 


296 


306 


317 


327 


337 


9 


.V 

9.0 


430 


347 


357 


367 


377 


387 


397 


407 


417 


428 


438 






431 


448 


458 


468 


478 


488 


498 


508 


518 


528 


538 




432 


548 


558 


568 


579 


589 


599 


609 


619 


629 


639 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 




434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 




485 


849 


859 


869 


879 


889 


899 


909 


919 


929 


939 




436 


949 


959 


969 


979 


988 


998 


*008 


*018 


*028 


*038 


9 


437 


64 048 


058 


068 


078 


088 


098 


108 


118 


128 


137 


1 


0.9 


438 


147 


157 


167 


177 


187 


197 


207 


217 


227 


237 


2 


1.8 


439 


246 


256 


266 


276 


286 


296 


306 


316 


326 


335 


3 


2.7 


440 


345 


355 


365 


375 


385 


395 


404 


414 


424 


434 


4 


3.6 


441 


444 


454 


464 


473 


483 


493 


503 


513 


523 


532 


5 


4.5 


442 


542 


552 


562 


572 


582 


591 


601 


611 


621 


631 


6 


5.4 


443 


640 


650 


660 


670 


680 


689 


699 


709 


719 


729 


7 


6.3 


444 


738 


748 


758 


768 


777 


787 


797 


807 


816 


826 


8 


7.2 


445 


836 


846 


856 


865 


875 


885 


895 


904 


914 


924 


9 


8.1 


446 


933 


943 


953 


963 


972 


982 


992 


*002 


*011 


*02I 




447 


65 031 


040 


050 


060 


070 


079 


089 


099 


108 


118 




448 


128 


137 


147 


157 


167 


176 


186 


196 


205 


215 




449 


225 


234 


244 


254 


263 


273 


283 


292 


302 


312 




450 


321 


331 


341 


350 


360 


369 


379 


389 


398 


408 




N. 


L. o 


1 ' 


3 


4 


5 


6 1 7 I 8 


9 


Prop. Parts 



24 



TABLE I 



450-500 



N. L. o i a | 3 








460 


65 321 


331 


341 


350 


360 


369 


379 


389 


398 


408 




451 


418 


427 


437 


447 


456 


466 


475 


485 


495 


504 




452 


514 


523 


533 


543 


552 


562 


571 


531 


591 


600 




453 


610 


619 


629 


639 


648 


658 


667 


677 


686 


696 




454 


706 


715 


725 


734 


744 


753 


763 


772 


782 


792 




456 


801 


811 


820 


830 


839 


849 


858 


868 


877 


887 




456 


896 


906 


916 


925 


935 


944 


954 


963 


973 


982 


10 


457 


992 


*001 


*011 


*020 


*030 


*039 


*049 


058 


*068 


*077 


1 


I.U 


458 


66 087 


096 


106 


115. 


124 


134 


143 


153 


162 


172 


2 


2 
3f\ 


459 


181 


191 


200 


210 


219 


229 


238 


247 


257 


266 


A 


.0 
A ft 


460 


276 


285 


295 


304 


314 


323 


332 


342 


351 


361 


f 

5 


V 

5 


461 


370 


380 


389 


398 


408 


417 


427 


436 


445 


455 


6 


6.0 


462 


464 


474 


483 


492 


502 


511 


521 


530 


539 


549 


7 


70 


463 


558 


567 


577 


586 


596 


605 


614 


624 


633 


642 


8 


s!o 


464 


652 


661 


671 


680 


689 


699 


708 


717 


727 


736 


9 




465 


745 


755 


764 


773 


783 


792 


801 


811 


820 


829 






466 


839 


848 


857 


867 


876 


885 


894 


904 


913 


922 




467 


932 


941 


950 


960 


969 


978 


987 


997 


*006 


*015 




468 


67 025 


034 


043 


052 


062 


071 


080 


089 


099 


108 




469 


117 


127 


136 


145 


154 


164 


'173 


182 


191 


201 




470 


210 


219 


228 


237 


247 


256 


265 


274 


284 


293 




471 


302 


311 


321 


330 


339 


348 


357 


367 


376 


385 


9 


472 


394 


403 


413 


422 


431 


440 


449 


459 


468 


477 


1 


U.V 


473 


486 


495 


504 


514 


523 


532 


541 


550 


560 


569 


2 


1.8 


474 


578 


587 


596 


605 


614 


624 


633 


642 


651 


660 


3 


2.7 


475 


669 


679 


688 


697 


706 


715 


724 


733 


742 


752 


4 


3.6 

4C 


476 


761 


770 


779 


788 


797 


806 


815 


825 


834 


843 




.J 

5 A 


477 


852 


861 


870 


879 


888 


897 


906 


916 


925 


934 




.T 
61 


478 


943 


952 


961 


970 


979 


988 


997 


*006 


*015 


*024 




. J 
7-1 


479 


68 034 


043 


052 


061 


070 


079 


088 


097 


106 


115 


q 


.L 
8. 1 


480 


124 


133 


142 


151 


160 


169 


178 


187 


196 


205 


t 




481 


215 


224 


233 


242 


251 


260 


269 


278 


287 


296 




482 


305 


314 


323 


332 


341 


350 


359 


368 


377 


386 




483 


395 


404 


413 


422 


431 


440 


449 


458 


467 


476 




484 


485 


494 


502 


511 


520 


529 


538 


547 


556 


565 




485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


655 




486 


664 


673 


681 


690 


699 


708 


717 


726 


735 


744 


8 


487 


753 


762 


771 


780 


789 


797 


806 


815 


824 


833 


1 


0.8 


488 


842 


851 


860 


869 


878 


886 


895 


904 


913 


922 


2 


1.6 


489 


931 


940 


949 


958 


966 


975 


984 


993 


*002 


*011 


3 


2.4 


490 


69 020 


028 


037 


046 


055 


064 


073 


082 


090 


099 


4 


3 2 


491 


108 


117 


126 


135 


144 


152 


161 


170 


179 


188 


5 


4.0 


492 


197 


205 


214 


223 


232 


241 


249 


258 


267 


276 


6 


4 8 


493 


285 


294 


302 


311 


320 


329 


338 


346 


355 


364 


7 


5 6 


494 


373 


381 


390 


399 


408 


417 


425 


434 


443 


452 


8 


6.4 


495 


461 


469 


478 


487 


496 


504 


513 


522 


531 


539 


9 


7.2 


496 


548 


557 


566 


574 


583 


592 


601 


609 


618 


627 




497 


636 


644 


653 


662 


671 


679 


688 


697 


705 


714 




498 


723 


732 


740 


749 


758 


767 


775 


784 


793 


801 




499 


810 


819 


827 


836 


845 


854 


862 


871 


880 


888 




600 


897 


906 


914 


923 


932 


940 


949 


958 


966 


975 




N. 


L. o 


I 2 


3 1 4 


5 


6 


7 


8 


9 


Prop. Parts 



25 



TABLE I 



500-550 



H. | 






















Prop 


Parts 


600 


69 897 


906 


914 


923 


932 


940 


949 


958 


966 


975 






501 


984 


992 


*001 


*010 


*018 


*027 


*036 


*044 


*053 


*062 






502 


70 070 


079 


088 


096 


105 


114 


122 


131 


140 


148 






503 


157 


165 


174 


183 


191 


200 


209 


217 


226 


234 






504 


243 


252 


260 


269 


278 


286 


295 


303 


312 


321 






605 


329 


338 


346 


355 


364 


372 


381 


389 


398 


406 






506 


415 


424 


432 


441 


449 


458 


467 


475 


484 


492 




9 


507 


501 


509 


518 


526 


535 


544 


552 


561 


569 


578 


1 


0.9 


508 


586 


595 


603 


612 


621 


629 


638 


646 


655 


663 


2 


1.8 

27 


509 


672 


680 


689 


697 


706 


714 


723 


731 


740 


749 


4 


.7 
3.6 


610 


757 


766 


774 


783 


791 


800 


808 


817 


825 


834 


5 


45 


511 


842 


851 


859 


868 


876 


885 


893 


902 


910 


919 


6 


5.4 


512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*003 


7 


63 


513 


71 012 


020 


029 


037 


046 


054 


063 


071 


079 


088 


8 


7 2 


514 


096 


105 


113 


122 


130 


139 


147 


155 


164 


172 


9 


8.1 


616 


181 


189 


198 


206 


214 


223 


231 


240 


248 


257 






516 


263 


273 


282 


290 


299 


307 


315 


324 


332 


341 






517 


349 


357 


366 


374 


383 


391 


399 


408 


416 


425 






518 


433 


441 


450 


458 


466 


475 


483 


492 


500 


508 






519 


517 


525 


533 


542 


550 


559 


567 


575 


584 


592 






620 


600 


609 


617 


625 


634 


642 


650 


659 


667 


675 






521 


684 


692 


700 


709 


717 


725 


734 


742 


750 


759 




8 


522 


767 


775 


784 


792 


800 


809 


817 


825 


834 


842 


1 


0.8 


523 


850 


858 


867 


875 


883 


892 


900 


908 


917 


925 


2 


1.6 


524 


933 


941 


950 


958 


966 


975 


983 


991 


999 


*008 


3 


2.4 


626 


72 016 


024 


032 


041 


049 


057 


066 


074 


082 


090 


4 


3.2 
4 O 


526 


099 


107 


115 


123 


132 


140 


148 


156 


165 


173 




.0 
40 


527 


181 


189 


198 


206 


214 


222 


230 


239 


247 


255 






5x 


528 


263 


272 


280 


288 


296 


304 


313 


321 


329 


337 






64 


529 


346 


354 


362 


370 


378 


387 


395 


403 


411 


419 


9 


A 
7.2 


630 


428 


436 


444 


452 


460 


469 


477 


485 


493 


501 






531 


509 


518 


526 


534 


542 


550 


558 


567 


575 


583 






532 


591 


599 


607 


616 


624 


632 


640 


648 


656 


665 






533 


673 


681 


689 


697 


705 


713 


722 


730 


738 


746 






534 


754 


762 


770 


779 


787 


795 


803 


811 


819 


827 






636 


835 


843 


852 


860 


868 


876 


884 


892 


900 


908 






536 


916 


925 


933 


941 


949 


957 


965 


973 


981 


989 




7 


537 


997 


*006 


*014 


*022 


*030 


*038 


*046 


*054 


*062 


*070 


1 


0.7 


538 


73 078 


086 


094 


102 


111 


119 


127 


135 


143 


151 


2 


1.4 


539 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 


3 


2 1 


640 


239 


247 


255 


263 


272 


280 


288 


296 


304 


312 


4 


2.8 


541 


320 


328 


336 


344 


352 


360 


368 


376 


384 


392 


5 


3 5 


542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


6 


4.2 


543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


7 


4.9 


544 


560 


568 


576 


584 


592 


600 


608 


616 


624 


632 


8 


5.6 


646 


640 


648 


656 


664 


672 


679 


687 


695 


703 


711 


9 


6.3 


546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 






547 


799 


807 


815 


823 


830 


838 


846 


854 


862 


870 






548 


878 


886 


894 


902 


910 


918 


926 


933 


941 


949 






549 


957 


965 


973 


981 


989 


997 


*005 


*013 


*020 


*028 






660 


74036 


044 


052 


060 


068 


076 


084 


092 


099 


107 






N. 


L. o 


z 


2 


3 


4 


5 


6 


7 


8 


9 


Prop 


. Parts 



26 



TABLE I 



550-600 



N. | L. Z 2 


3 \ 4 


5 | 6 7 | 8 9 


Prop. Ptrtt 


660 


74 036 


044 


052 


060 


068 


076 


084 


092 


099 


107 






551 


115 


123 


131 


139 


147 


155 


162 


170 


178 


186 






552 


194 


202 


210 


218 


225 


233 


241 


249 


257 


265 






553 


273 


280 


288 


296 


304 


312 


320 


327 


335 


343 






554 


351 


359 


367 


374 


382 


390 


398 


406 


414 


421 






656 


429 


437 


445 


453 


461 


468 


476 


484 


492 


500 






556 


507 


515 


523 


531 


539 


547 


554 


562 


570 


578 






557 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 






558 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 






559 


741 


749 


757 


764 


772 


780 


788 


796 


803 


811 






660 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 






561 


896 


904 


912 


920 


927 


935 


943 


950 


958 


966 


1 


0.8 


562 


974 


981 


989 


997 


*005 


*012 


*020 


*028 


*035 


*043 


2 


1.6 


563 


75 051 


059 


066 


074 


082 


089 


097 


105 


113 


120 


3 


24 


564 


128 


136 


143 


151 


159 


166 


174 


182 


189 


197 


4 


32 


666 


205 


213 


220 


228 


236 


243 


251 


259 


266 


274 


5 


4.0 


566 


282 


289 


297 


305 


312 


320 


328 


335 


343 


351 


6 


4.8 


567 


358 


366 


374 


381 


389 


397 


404 


412 


420 


427 


7 


5.6 


568 


435 


442 


450 


458 


465 


473 


481 


488 


496 


504 


8 


6.4 


569 


511 


519 


526 


534 


542 


549 


557 


565 


572 


580 


9 


7.2 


670 


587 


595 


603 


610 


618 


626 


633 


641 


648 


656 






571 


664 


671 


679 


686 


694 


702 


709 


717 


724 


732 






572 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 






573 


815 


823 


831 


838 


846 


853 


861 


868 


876 


884 






574 


891 


899 


906 


914 


921 


929 


937 


944 


952 


959 






676 


967 


974 


982 


989 


997 


*005 


*012 


*020 


*027 


*035 






576 


76 042 


050 


057 


065 


072 


080 


087 


095 


103 


110 






577 


118 


125 


133 


140 


148 


155 


163 


170 


178 


185 






578 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 






579 


268 


275 


283 


290 


298 


305 


313 


320 


328 


335 






680 


343 


350 


358 


365 


373 


380 


388 


395 


403 


410 




7 


581 


418 


425 


433 


440 


448 


455 


462 


470 


477 


485 


1 


7 


582 


492 


500 


507 


515 


522 


530 


537 


545 


552 


559 


2 


V . 1 

1 4 


583 


567 


574 


582 


589 


597 


604 


612 


619 


626 


634 


3 


2 1 


584 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 


4 


2 8 


686 


716 


723 


730 


738 


745 


753 


760 


768 


775 


782 


5 


3 5 


586 


790 


797 


805 


812 


819 


827 


834 


842 


849 


856 


6 


' 4 2 


587 


864 


871 


879 


886 


893 


901 


908 


916 


923 


930 


7 


4.9 


588 


938 


945 


953 


960 


967 


975 


982 


989 


997 


*004 


8 


5.6 


589 


77 012 


019 


026 


034 


041 


048 


056 


063 


070 


078 


9 


6.3 


690 


085 


093 


100 


107 


115 


122 


129 


137 


144 


151 






591 


159 


166 


173 


181 


188 


195 


203 


210 


217 


225 






592 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 






593 


305 


313 


320 


327 


335 


342 


349 


357 


364 


371 






594 


379 


386 


393 


401 


408 


415 


422 


430 


437 


444 






696 


452 


459 


466 


474 


481 


488 


495 


503 


510 


517 






596 


525 


532 


539 


546 


554 


561 


568 


576 


583 


590 






597 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 






598 


670 


677 


685 


692 


699 


706 


714 


721 


728 


735 






599 


743 


750 


757 


764 


772 


779 


786 


793 


801 


808 






600 


815 


822 


830 


837 


844 


851 


859 


866 


873 


880 






If. 


L. o i 


i 


3 


4 


s 


6 


7 


8 


9 


Prop. Parts 



27 



TABLE I 



600-660 



N. 






















Prof 


. Parts 


600 


77 815 


822 


830 


837 


844 


851 


859 


866 


873 


880 






601 


887 


895 


902 


909 


916 


924 


931 


938 


945 


952 






602 


960 


967 


974 


981 


988 


996 


*003 


*010 


*017 


*025 






603 


78 032 


039 


046 


053 


061 


068 


075 


082 


089 


097 






604 


104 


111 


118 


125 


132 


140 


147 


154 


161 


168 






605 


176 


183 


190 


197 


204 


211 


219 


226 


233 


240 






606 


247 


254 


262 


269 


276 


283 


290 


297 


305 


312 




OQ 


607 


319 


326 


333 


340 


347 


355 


362 


369 


376 


383 




.8 
1^ 


608 


390 


398 


405 


412 


419 


426 


433 


440 


447 


455 




.0 

2 A 


609 


462 


469 


476 


483 


490 


497 


504 


512 


519 


526 


4 


,*r 

3 2 


610 


533 


540 


547 


554 


561 


569 


576 


583 


590 


597 


5 


4*0 


611 


604 


611 


618 


625 


633 


640 


647 


654 


661 


668 


6 


48 


612 


675 


682 


689 


696 


704 


711 


718 


725 


732 


739 


7 


5^6 


613 


746 


753 


760 


767 


774 


781 


789 


796 


803 


810 


g 


64 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


9 


l.l 


615 


888 


895 


902 


909 


916 


923 


930 


937 


944 


951 






616 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 






617 


79 029 


036 


043 


050 


057 


064 


071 


078 


085 


092 






618 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 






619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 






620 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 






621 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 




7 


622 


379 


386 


393 


400 


407 


414 


421 


428 


43.) 


442 


1 


7 


623 


449 


456 


463 


470 


477 


484 


491 


498 


505 


511 


2 


1.4 


624 


518 


525 


532 


539 


546 


553 


560 


567 


574 


581 


3 


2.1 


625 


588 


595 


602 


609 


616 


623 


630 


637 


644 


650 


4 


2.8 

3C 


626 


657 


664 


671 


678 


685 


692 


699 


706 


713 


720 




. J 

40 


627 


727 


734 


741 


748 


754 


761 


768 


775 


782 


789 




.Z 
4O 


628 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 




.7 

5i 


629 


865 


872 


879 


886 


893 


900 


906 


913 


920 


927 


9 


.0 

6.3 


630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 






631 


80 003 


010 


017 


024 


030 


037 


044 


051 


058 


065 






632 


072 


079 


085 


092 


099 


106 


113 


120 


127 


134 






633 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 






634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 






635 


277 


284 


291 


298 


305 


312 


318 


325 


332 


339 






636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 




6 


637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 


1 


0.6 


638 


482 


489 


496 


502 


509 


516 


523 


530 


536 


543 


2 


1.2 


639 


550 


557 


564 


570 


577 


584 


591 


598 


604 


611 


3 


1.8 


640 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


4 


2.4 


641 


686 


693 


699 


706 


713 


720 


726 


733 


740 


747 


5 


3.0 


642 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


6 


3.6 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 


7 


4.2 


644 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 


8 


4.8 


645 


956 


963 


969 


976 


983 


990 


996 


*003 


*010 


*017 


9 


5.4 


646 


81 023 


030 


037 


043 


050 


057 


064 


070 


077 


084 






647 


090 


097 


104 


111 


117 


124 


131 


137 


144 


151 






648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 






649 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 






660 


291 


298 


305 


311 


318 


325 


331 


338 


345 


351 






V. 


L. o 


< 


2 


3 


4 


s 


6 


7 


8 


9 


Prof 


. Parts 



28 



TABLE I 



650-700 



N. 


L. 1 9 3 4 


5 6 7 8 9 | Prop. Parts 


660 


81 291 


298 


305 


311 


318 


325 


331 


338 


345 


351 






651 


358 


365 


371 


378 


385 


391 


398 


405 


411 


418 






652 


425 


431 


438 


445 


451 


458 


465 


471 


478 


485 






653 


491 


498 


505 


511 


518 


525 


531 


538 


544 


551 






654 


558 


564 


571 


578 


584 


591 


598 


604 


611 


617 






655 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 






656 


690 


697 


704 


710 


717 


723 


730 


737 


743 


750 






657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 






658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 






659 


889 


895 


902 


908 


915 


921 


928 


935 


941 


948 






660 


954 


961 


968 


974 


981 


987 


994 


*000 


*007 


*014 






661 


82 020 


027 


033 


040 


046 


053 


060 


066 


073 


079 




7 


662 


086 


092 


099 


105 


112 


119 


125 


132 


138 


145 


1 


0.7 


663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


2 


1 4 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


3 


2 1 


665 


282 


289 


295 


302 


308 


315 


321 


328 


334 


34! 


4 


2 8 

3c 


666 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 




5 
4<) 


667 


413 


419 


426 


432 


439 


445 


452 


458 


465 


471 




L 

4Q 


668 


478 


484 


491 


497 


504 


510 


517 


523 


530 


536 




.V 
5f 


669 


543 


549 


556 


562 


569 


575 


582 


588 


595 


601 


9 


.0 

6.3 


670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 






671 


672 


679 


685 


692 


698 


705 


711 


718 


724 


730 






672 


737 


743 


750 


756 


763 


769 


776 


782 


789 


795 






673 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 






674 


866 


872 


879 


885 


892 


898 


905 


911 


918 


924 






675 


930 


937 


943 


950 


956 


963 


969 


975 


982 


988 






676 


995 


*001 


*008 


*014 


*020 


*027 


*033 


*040 


*046 


*052 






677 


83 059 


065 


072 


078 


085 


091 


097 


104 


110 


117 






678 


123 


129 


136 


142 


149 


155 


161 


168 


174 


181 






679 


187 


193 


200 


206 


213 


219 


225 


232 


238 


245 






680 


251 


257 


264 


270 


276 


283 


289 


296 


302 


308 






681 


315 


321 


327 


334 


340 


347 


353 


359 


366 


372 




6 


682 


378 


385 


391 


398 


404 


410 


417 


423 


429 


436 


1 


6 


683 


442 


448 


455 


461 


467 


474 


480 


487 


493 


499 


2 


1.2 


684 


506 


512 


518 


525 


531 


537 


544 


550 


556 


563 


3 


1.8 


686 


569 


575 


582 


588 


594 


601 


607 


613 


620 


626 


4 


2.4 
3/% 


686 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 




.0 
3f 


687 


696 


702 


708 


715 


721 


727 


734 


740 


746 


753 




.6 
4n 


688 
689 


759 

822 


765 
828 


771 
835 


778 
841 


784 
847 


790 
853 


797 
860 


803 
866 


809 
872 


816 
879 


8 
9 


L 
4.8 
5.4 


690 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 






691 


948 


954 


960 


967 


973 


979 


985 


992 


998 


*004 






692 


84 Oil 


017 


023 


029 


036 


042 


048 


055 


061 


067 






693 


073 


080 


086 


092 


098 


105 


111 


117 


123 


130 






694 


136 


142 


148 


155 


161 


167 


173 


180 


186 


192 






695 


198 


205 


211 


217 


223 


230 


236 


242 


248 


255 






696 


261 


267 


273 


280 


286 


292 


298 


305 


311 


317 






697 


323 


330 


336 


342 


348 


354 


361 


367 


373 


379 






698 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 






699 


448 


454 


460 


466 


473 


479 


485 


491 


497 


504 






700 


510 


516 


522 


528 


535 


541 


547 


553 


559 


566 






N. 


L. o 


i 


a 


3 | 4 


S 


6 


7 


8 9 


Prop. Parts 



TABLE I 



700-750 



w. 


L. o 


i 


2 


3 


4 


5 


6 


7 


8 


9 




700 


84 510 


516 


522 


528 


535 


541 


547 


553 


559 


566 




701 


572 


578 


584 


590 


597 


603 


609 


615 


621 


628 




702 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 




703 


696 


702 


708 


714 


720 


726 


733 


739 


745 


751 




704 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 




705 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 


7 


706 


880 


887 


893 


899 


905 


911 


917 


924 


930 


936 


1 




n 7 


707 


942 


948 


954 


960 


967 


973 


979 


985 


991 


997 


1 
7 


u / 
1 4 


708 


85 003 


009 


016 


022 


028 


034 


040 


046 


052 


058 


L 
3 


1 T 

7 1 


709 


065 


071 


077 


083 


089 


095 


101 


107 


114 


120 


J 

4 


L \ 

2.8 


710 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 


5 


3.5 


711 


187 


193 


199 


205 


211 


217 


224 


230 


236 


242 


6 


4.2 


712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 


7 


4 9 


713 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 


8 


56 


714 


370 


376 


382 


388 


394 


400 


406 


412 


418 


425 


9 


6J 


715 


431 


437 


443 


449 


455 


461 


467 


473 


479 


485 




716 


491 


497 


503 


509 


516 


522 


528 


534 


540 


546 




717 


552 


558 


564 


570 


576 


582 


588 


594 


600 


606 




718 


612 


618 


625 


631 


637 


643 


649 


655 


661 


667 




719 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 




720 


733 


739 


745 


751 


757 


763 


769 


775 


781 


788 




721 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 


6 


722 


854 


860 


866 


872 


878 


884 


890 


896 


902 


908 


I 


0.6 


723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


2 


1.2 


724 


974 


980 


986 


992 


998 


*004 


*010 


*016 


*022 


*028 


3 


1.8 


725 


86 034 


040 


046 


052 


058 


064 


070 


076 


082 


088 


4 


2.4 
3/\ 


726 


094 


100 


106 


112 


118 


124 


130 


136 


141 


147 




.0 

3c 


727 


153 


159 


165 


171 


177 


183 


189 


195 


201 


207 




.0 

4-5 


728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 




2 
40 


729 


273 


279 


285 


291 


297 


303 


308 


314 


320 


326 


8 
9 




5 4 


730 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 






731 


392 


398 


404 


410 


415 


421 


427 


433 


439 


445 




732 


451 


457 


463 


469 


475 


481 


487 


493 


499 


504 




733 


510 


516 


522 


528 


534 


540 


546 


552 


558 


564 




734 


570 


576 


581 


587 


593 


599 


605 


611 


617 


623 




735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 




736 


688 


694 


700 


705 


711 


717 


723 


729 


735 


741 


6 


737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 


1 


0.5 


738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


2 


1.0 


739 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


3 


1 5 


740 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 


4 


2 


741 


982 


988 


994 


999 


*005 


*OI1 


*017 


*023 


*029 


*035 


5 


2 5 


742 


87 040 


046 


052 


058 


064 


070 


075 


081 


087 


093 


6 


3 


743 


099 


105 


111 


116 


122 


128 


134 


140 


146 


151 


7 


3 5 


744 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 


8 


4 


745 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 


9 


4 5 


746 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 




747 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 




748 


390 


396 


402 


408 


413 


419 


425 


431 


437 


442 




749 


448 


454 


460 


466 


471 


477 


483 


489 


495 


500 




750 


506 


512 


518 


523 


529 


535 


541 


547 


552 


558 




tt. 


L. o 


z 


a 


3 


4 


5 


6 


7 


8 


9 


Prop. Parts 



TABLE I 



750-800 



N. 


L. o 


X 


ai 


3 


4 


5 


6 


7 


8 


9 


Prop* P&rts 


760 


87 506 


512 


518 


523 


529 


535 


541 


547 


552 


558 




751 


564 


570 


576 


581 


587 


593 


599 


604 


610 


616 




752 


622 


628 


633 


639 


645 


651 


656 


662 


668 


674 




753 


679 


685 


691 


697 


703 


708 


714 


720 


726 


731 




754 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




755 


793 


800 


806 


812 


818 


823 


829 


835 


841 


846 




756 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




757 


910 


915 


921 


927 


933 


938 


944 


950. 


935 


961 




758 


967 


973 


978 


904 


990 


996 


*OJ1 


*007 


013 


018 




759 


88024 


030 


036 


041 


047 


053 


058 


064 


0/0 


076 




760 


081 


087 


093 


098 


104 


110 


116 


121 


127 


133 




761 


13 


H4 


150 


156 


161 


167 


173 


17,8 


184 


190 


i t\ a 


762 


195 


201 


207 


213 


218 


224 


2W 


235 


241 


247 


1 U.o 

9 1 ? 


763 


252 


258 


264 


270 


275 


281 


287 


292 


298 


304 


L \ .it 
31 ft 


764 


309 


315 


321 


326 


332 


338 


343 


349 


355 


360 


1 .0 

4 2.4 


766 


366 


372 


377 


383 


389 


395 


400 


406 


412 


417 


5 3.0 


766 


423 


429 


434 


440 


446 


451 


457 


463 


468 


474 


6 3.6 


767 


480 


485 


491 


497 


502 


508 


513 


519 


525 


530 


742 


768 


536 


542 


547 


553 


559 


564 


570 


576 


581 


587 


8 4.8 


769 


593 


598 


604 


610 


615 


621 


627 


632 


638 


643 


9 5.4 


770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 




771 


705 


711 


717 


722 


728 


734 


739 


745 


750 


756 




772 


762 


767 


773 


779 


704 


790 


795 


801 


807 


812 




773 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 




774 


874 


880 


885 


891 


897 


902 


908 


913 


919 


925 




775 


930 


936 


941 


947 


953 


958 


964 


969 


975 


981 




776 


986 


992 


997 


003 


009 


*014 


*020 


*025 


031 


037 




777 


89 042 


048 


053 


059 


064 


070 


076 


081 


087 


092 




778 


098 


104 


109 


115 


120 


126 


131 


137 


143 


148 




779 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 




780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 




781 


265 


271 


276 


282 


207 


293 


298 


304 


310 


315 


5 


782 


321 


326 


332 


337 


343 


348 


354 


360 


365 


371 


1 0.5 


783 


376 


382 


387 


393 


398 


404 


409 


415 


421 


426 


2 1.0 


784 


432 


437 


443 


448 


454 


459 


465 


470 


476 


481 


3 1.5 


786 


487 


492 


498 


504 


509 


515 


520 


526 


531 


537 


4 2.0 

c 9 e 


786 


542 


548 


553 


559 


564 


570 


575 


581 


586 


592 


j L.J 
61 A 


787 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 


J . v 
73 r 


788 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 


J.J 

84 ft 


789 


708 


713 


719 


724 


730 


735 


741 


746 


752 


757 


T . U 

9 4.5 


790 


763 


768 


774 


779 


785 


790 


796 


801 


807 


812 




791 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 




792 


873 


878 


883 


889 


894 


900 


905 


911 


916 


922 




793 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




794 


982 


988 


993 


998 


*004 


*009 


*015 


*020 


*026 


*031 




795 


90 037 


042 


048 


053 


059 


064 


069 


073 


080 


086 




796 


091 


097 


102 


108 


113 


119 


124 


129 


135 


140 




797 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 




798 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 




799 


255 


260 


266 


271 


276 


282 


287 


293 


298 


304 




800 


309 


314 


320 


325 


331 


336 


342 


347 


352 


358 




W. 


L. o 


X 


a 


3 


4 


5 


6 


7 


8 


9 


Prop. Parta 



31 



TABLE I 



800-850 



w. 


L. | 


X 


a 


3 












9 


Prop. 


Parts 


800 


90 309 


314 


320 


325 


331 


336 


342 


347 


352 


358 






801 


363 


369 


374 


380 


385 


390 


396 


401 


407 


412 






802 


417 


423 


428 


434 


439 


445 


450 


455 


461 


466 






803 


472 


477 


482 


488 


493 


499 


504 


509 


515 


520 






804 


526 


531 


536 


542 


547 


553 


558 


563 


569 


574 






805 


580 


585 


590 


596 


601 


607 


612 


617 


623 


628 






806 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 






807 


687 


693 


698 


703 


709 


714 


720 


725 


730 


736 






808 


741 


747 


752 


757 


763 


768 


773 


779 


784 


789 






809 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 






810 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 






811 


902 


907 


913 


918 


924 


929 


934 


940 


945 


950 




6 


812 


956 


961 


966 


972 


977 


902 


908 


993 


998 


*004 


1 


0.6 


813 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


057 


2 


1.2 


814 


062 


068 


073 


078 


084 


089 


094 


100 


105 


110 


3 


1 8 


816 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 


4 

c 


2 4 
7 n 


816 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 


J 


J U 
1 A 


817 


III 


228 


233 


238 


243 


249 


254 


259 


265 


270 


7 


J O 

4 2 


818 


275 


281 


206 


291 


297 


302 


307 


312 


318 


323 




A a 


819 


328 


334 


339 


344 


350 


355 


360 


365 


371 


376 


9 


*F . O 

5.4 


820 


381 


387 


392 


397 


403 


408 


413 


418 


424 


429 






821 


434 


440 


445 


450 


455 


461 


466 


471 


477 


482 






822 


487 


492 


498 


503 


508 


514 


519 


524 


529 


535 






823 


540 


545 


551 


556 


561 


566 


572 


577 


582 


587 






824 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 






825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


693 






826 


698 


703 


7C9 


714 


719 


724 


730 


735 


740 


745 






827 


751 


756 


761 


766 


772 


777 


782 


787 


793 


798 






828 


803 


808 


814 


819 


824 


829 


834 


840 


845 


850 






829 


855 


861 


866 


871 


876 


882 


887 


892 


897 


903 






830 


908 


913 


918 


924 


929 


934 


939 


944 


950 


955 






831 


960 


965 


971 


976 


981 


906 


991 


997 


*602 


*007 




5 


832 


92 012 


018 


023 


028 


033 


038 


044 


049 


054 


059 


1 


5 


833 


065 


070 


075 


OCO 


OC5 


091 


096 


101 


106 


111 


2 


1 


834 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 


3 


1 5 


835 


169 


174 


179 


184 


189 


195 


200 


205 


210 


215 


4 


2 


836 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 


5 


2 5 


837 


273 


278 


283 


288 


293 


298 


304 


309 


314 


319 


6 


3.0 


838 


324 


330 


335 


340 


345 


350 


355 


361 


366 


371 


7 


3.5 


839 


376 


381 


387 


392 


397 


402 


407 


412 


418 


423 


8 

Q 


4.0 
4 < 


840 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 


7 


~. j 


841 


480 


485 


490 


495 


500 


505 


511 


516 


521 


526 






842 


531 


536 


542 


547 


552 


557 


562 


567 


572 


578 






843 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 






844 


634 


639 


645 


650 


655 


660 


665 


670 


675 


681 






846 


686 


691 


696 


701 


706 


711 


716 


722 


727 


732 






846 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 






847 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 






848 


840 


845 


850 


855 


860 


865 


870 


875 


881 


886 






849 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 






860 


942 


947 


952 


957 


962 


967 


973 


978 


983 


988 






N. 


L. o 


I 


a 


3 


4 


S 


1 * 


7 


8 


9 


Prop. 


Parts 



32 



TABLE I 



850-900 



N. 


L. 


X 


a 


3 4 


5 6 7 


8 9 


Prop* Pfttts 


860 


92 942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




851 


993 


998 


003 


008 


"013 


018 


024 


*029 


034 


039 




852 


93 044 


049 


054 


059 


064 


069 


075 


080 


085 


090 




853 


095 


100 


105 


110 


115 


120 


125 


131 


136 


141 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




865 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 





856 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 




o 

Ot 


857 


298 


303 


308 


313 


318 


323 


328 


334 


339 


344 




o 

1 


858 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 




1 .1 

10 


859 


399 


404 


409 


414 


420 


425 


430 


435 


440 


445 


4 


.0 

2 4 


860 


450 


455 


460 


463 


470 


475 


480 


485 


490 


495 


5 


3.0 


861 


500 


505 


510 


515 


520 


526 


531 


536 


541 


546 


6 


3.6 


862 


551 


556 


561 


566 


571 


576 


581 


586 


591 


596 


7 


4.2 


863 


601 


606 


611 


616 


621 


626 


631 


636 


641 


646 


8 


4 8 


864 


651 


656 


661 


666 


671 


676 


6*2 


687 


692 


697 


9 


5.4 


865 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




869 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 


94 002 


007 


012 


017 


022 


027 


032 


037 


042 


047 


6 


872 


052 


057 


062 


067 


072 


077 


082 


086 


091 


096 


1 


U.D 


873 


101 


106 


111 


116 


121 


126 


131 


136 


141 


146 


2 


1.0 


874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 


3 


1.5 
2r\ 


875 


201 


206 


211 


216 


221 


226 


231 


236 


240 


245 


c 


.0 

2c 


876 


250 


255 


260 


265 


270 


275 


280 


285 


290 


295 


J 


.j 


877 


300 


305 


310 


315 


320 


325 


330 


335 


340 


345 


7 


~'e 


878 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


/ 


4 fl 


879 


399 


404 


409 


414 


419 


424 


429 


433 


438 


443 


9 


*f . U 

4.5 


880 


448 


453 


458 


463 


468 


473 


478 


483 


488 


493 






881 


498 


503 


507 


512 


517 


522 


527 


532 


537 


542 




882 


547 


552 


557 


562 


567 


571 


576 


581 


586 


591 




883 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 




884 


645 


650 


655 


660 


665 


670 


675 


680 


685 


689 




885 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 




886 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 


4 


887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 


1 


0.4 


888 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 


2 


0.8 


889 


890 


895 


900 


905 


910 


915 


919 


924 


929 


934 


3 


1.2 


890 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 


4 


1.6 


891 


988 


093 


998 


*002 


*007 


*012 


*017 


*022 


*027 


*032 


5 


2.0 


892 


95 036 


041 


046 


051 


056 


061 


066 


071 


075 


080 


6 


2.4 


893 


085 


090 


095 


100 


105 


109 


114 


119 


124 


129 


7 


2.8 


894 


134 


139 


143 


148 


153 


158 


163 


168 


173 


177 


8 


3.2 


895 


182 


187 


192 


197 


202 


207 


211 


216 


221 


226 







896 


231 


236 


240 


245 


250 


255 


260 


265 


270 


y4 




897 


279 


284 


289 


294 


299 


303 


308 


313 


318 


323 




898 


328 


332 


337 


342 


347 


352 


357 


361 


366 


371 




899 


376 


381 


386 


390 


395 


400 


405 


410 


415 


419 




900 


424 


429 


434 


439 


444 


448 


453 


458 


463 


468 




N. 


J^L. o 


X 


a 


3 


4 


5 


6 


r 


8 


9 


Prop. Parts 



33 



TABLE I 



900-950 



N. L. o x a 3 


4 5 6 


7 


8 


9 Prop. Parts 


900 


95424 


429 


434 


439 


444 


448 


453 


458 


463 


468 






901 


472 


477 


482 


487 


492 


497 


501 


506 


511 


516 






902 


521 


525 


530 


535 


540 


545 


550 


554 


559 


564 






903 


569 


574 


578 


583 


588 


593 


598 


602 


607 


612 






904 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 






905 


663 


670 


674 


679 


684 


689 


694 


698 


703 


708 






906 


713 


718 


722 


727 


732 


737 


742 


746 


751 


756 






907 


761 


766 


770 


775 


780 


785 


789 


794 


799 


804 






908 


809 


813 


818 


823 


828 


832 


837 


842 


847 


852 






909 


856 


861 


866 


871 


875 


880 


885 


890 


895 


899 






910 


904 


909 


914 


918 


923 


928 


933 


938 


942 


947 






911 


952 


957 


961 


966 


971 


976 


980 


985 


990 


995 




5 


912 


999 


*004 


*009 


*014 


*019 


*023 


*028 


*033 


*038 


*042 


1 


0.5 


913 


96 047 


052 


057 


061 


066 


071 


076 


080 


085 


090 


2 


1 


914 


095 


099 


104 


109 


114 


118 


123 


128 


133 


137 


3 


1.5 


915 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 


4 

c 


2.0 

9 "> 


916 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 


J 


L . J 

a n 


917 


237 


242 


246 


251 


256 


261 


265 


270 


275 


280 


7 


J U 

7 e 


918 


284 


289 


294 


298 


303 


308 


313 


317 


322 


327 


/ 


. J 

4 n 


919 


332 


336 


341 


346 


350 


355 


360 


365 


369 


374 


9 


T . V 

4 5 


920 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 






921 


426 


431 


435 


440 


445 


450 


454 


459 


464 


468 






922 


473 


478 


483 


487 


492 


497 


501 


506 


511 


515 






923 


520 


525 


530 


534 


539 


544 


548 


553 


558 


562 






924 


567 


572 


577 


581 


586 


591 


595 


600 


605 


609 






925 


614 


619 


624 


628 


633 


638 


642 


647 


652 


656 






926 


661 


666 


670 


675 


680 


685 


689 


694 


699 


703 






927 


708 


713 


717 


722 


727 


731 


736 


741 


745 


750 






928 


755 


759 


764 


769 


774 


778 


783 


788 


792 


797 






929 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 






930 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 






931 


895 


900 


904 


909 


914 


918 


923 


928 


932 


937 




4 


932 


942 


946 


951 


956 


960 


965 


970 


974 


979 


984 


1 


0.4 


933 


988 


993 


997 


*002 


*007 


*011 


*016 


*021 


*025 


*030 


2 


0.8 


934 


97 035 


039 


044 


049 


053 


058 


063 


067 


072 


077 


3 


1.2 


935 


081 


086 


090 


095 


100 


104 


109 


114 


118 


123 


4 


1.6 


936 


128 


132 


137 


142 


146 


151 


155 


160 


165 


169 


5 


2.0 


937 


174 


179 


183 


188 


192 


197 


202 


206 


211 


216 


6 


2.4 


938 


220 


225 


230 


234 


239 


243 


248 


253 


257 


262 


7 


2.8 


939 


267 


271 


276 


280 


285 


290 


294 


299 


304 


308 


8 


3.2 
3t 


940 


313 


317 


322 


327 


331 


336 


340 


345 


350 


354 


9 


.6 


941 


359 


364 


368 


373 


377 


382 


387 


391 


396 


400 






942 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 






943 


451 


456 


460 


465 


470 


474 


479 


483 


488 


493 






944 


497 


502 


506 


511 


516 


520 


525 


529 


534 


539 






945 


543 


548 


552 


557 


562 


566 


571 


575 


580 


585 






946 


589 


594 


598 


603 


607 


612 


617 


621 


626 


630 






947 


635 


640 


644 


649 


653 


658 


663 


667 


672 


676 






948 


681 


685 


690 


695 


699 


704 


708 


713 


717 


722 






949 


727 


731 


736 


740 


745 


749 


754 


759 


763 


768 






960 


772 


777 


782 


786 


791 


795 


800 


804 


809 


813 






N. 


L. o | x 


a 


3 


4 


5 


6 


r 1 8 


9 


Prop* Parts 



34 



TABLE I 



950-1000 



N. | L. o i a 3 4 5 6 7 8 


9 


Prop. Parts 


960 


97 772 


777 


782 


786 


791 


795 


800 


804 


809 


813 






951 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 






952 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 






953 


909 


914 


918 


923 


928 


932 


937 


941 


946 


950 






954 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 






955 


98 000 


005 


009 


014 


019 


023 


028 


032 


037 


041 






956 


046 


050 


055 


059 


064 


068 


073 


078 


082 


087 






957 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 






958 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 






959 


182 


186 


191 


195 


200 


204 


209 


214 


218 


223 






960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 




5 


961 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 


1 


5 


962 


318 


322 


327 


331 


336 


340 


345 


?49 


354 


358 


i 
2 


V . J 

1 


963 


363 


367 


372 


376 


381 


385 


390 


394 


399 


403 


3 


1 . V 

\ 5 


964 


408 


412 


417 


421 


426 


430 


435 


439 


444 


448 


4 


2!o 


965 


453 


457 


462 


466 


471 


475 


480 


484 


489 


493 


5 


2.5 


966 


498 


502 


507 


511 


516 


520 


525 


529 


534 


538 


6 


3.0 


967 


543 


547 


552 


556 


561 


565 


570 


574 


579 


583 


7 


3.5 


968 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


8 


4.0 


969 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 


9 


4.5 


970 


677 


682 


686 


691 


695 


700 


704 


709 


713 


717 






971 


722 


726 


731 


735 


740 


744 


749 


753 


758 


762 






972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 






973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 






974 


856 


860 


865 


869 


874 


873 


883 


887 


892 


896 






975 


900 


905 


909 


914 


918 


923 


927 


932 


936 


941 






976 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 






977 


989 


994 


998 


*003 


"007 


*012 


*016 


*021 


*025 


*029 






978 


99 034 


038 


043 


047 


052 


056 


061 


065 


069 


074 






979 


078 


083 


087 


092 


096 


100 


105 


109 


114 


118 






980 


123 


127 


131 


136 


140 


145 


149 


154 


158 


162 




4 


981 


167 


171 


176 


180 


185 


189 


193 


198 


202 


207 


\ 


4 


982 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 


2 


08 


983 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 


3 


1 2 


984 


300 


304 


308 


313 


317 


322 


326 


330 


335 


339 


4 


L6 


985 


344 


348 


352 


357 


361 


366 


370 


374 


379 


383 


5 


2 


986 


388 


392 


396 


401 


405 


410 


414 


419 


423 


427 


6 


2.4 


987 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 


7 


2.8 


988 


476 


480 


484 


489 


493 


498 


502 


506 


511 


515 


8 


3.2 


989 


520 


524 


528 


533 


537 


542 


546 


550 


555 


559 


9 


3.6 


990 


564 


568 


572 


577 


581 


585 


590 


594 


599 


603 






991 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 






992 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 






993 


695 


699 


704 


708 


712 


717 


721 


726 


730 


734 






994 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 






995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 






996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 






997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 






998 


913 


917 


922 


926 


930 


935 


939 


944 


948 


952 






999 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 






1000 


00 000 


004 


009 


013 


017 


022 


026 


030 


035 


039 






H. 


L. o 


z 


2 


3 


4 


5 


6 


7 


8 


9 


Prop. Parts 



TABLE II 
LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 







TABLK II 



179 r 




60 
120 
180 
240 


' 


1 sin 


log 5 


1 <!8tt 

Infinite! 
13.53627 
23524 
05915 
12.93421 


I tan 


log T 


fcot 

Infinite. 
13.53627 
23524 
05915 
12.93421 


I HOC 

00000 
00000 
00000 
00000 
00000 


I COH 


60 

59 

58 
57 
56 




1 
2 
3 

4 


Inf. neg. 
6.46373 
76476 
94085 
7.06579 


5.31443 
5.31443 
5.31443 
5.31443 


Inf. neg. 
6 46373 
76476 
94085 
7 06579 


5 31 443 
5.31443 
5.31443 
5 31 442 


10 OOOOC 
00000 
00000 
OOOOC 
00000 


300 
360 
420 
480 
540 


5 

6 
7 

8 
9 


7.16270 
24188 
30882 
36682 
41797 
7.46373 
50512 
54291 
57767 
60985 


5.31443 
5.31443 
5.31443 
5 31 443 
5.31443 


12 83730 
75812 
69118 
63318 
58203 


7.16270 
24188 
30882 
36682 
41797 


5.31442 
5.31442 
5.31442 
5.31442 
5.31442 


12.83730 
75812 
69118 
63318 
58203 


0.00000 
00000 
00000 
00000 
00000 


10.00000 
00000 
00000 
00000 
00000 


55 

54 
53 
52 
51 


600 
660 
720 
780 
840 
900 
960 
1020 
1080 
1140 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


5.31443 
5.31443 
5.31443 
5.31443 
5.31443 
5.31443 
5 31 443 
5.31443 
5.31443 
5.31443 


12.53627 
49488 
45709 
42233 
39015 


7.46373 
50512 
54291 
57767 
60986 
7 63982 
66785 
69418 
71900 
74248 


5.31442 
5 31 442 
5 31 442 
5.31442 
5.31442 


12.53627 
49488 
45709 
42233 
39014 


0.00000 
00000 
00000 
00000 
00000 


10.00000 
00000 
00000 
00000 
00000 


50 

49 
48 
47 
46 
45 
44 
43 
42 
41 


7.63982 
66784 
69417 
71900 
74248 


12.36018 
33216 
30583 
28100 
25752 


5 31 442 
5.31442 
5 31 442 
5 31 442 
5 31 442 
5 31 442 
5.31442 
5 31 442 
5 31 442 
5 31 442 


12 3C018 
33215 
30582 
28100 
25752 


10 00000 
00000 
00001 
00001 
00001 


10 00000 
00000 
9 99999 
99999 
99999 


1200 
1260 
1320 
1380 
1440 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


7 76475 
78594 
80615 
82545 
84393 


fl 31 443 
5 31 443 
5 31443 
5 31 443 
5.31443 
531 443 
5 31 443 
5 31 443 
5 31 443 
5 31443 


12 23525 
21406 
19385 
17455 
15607 


7.76476 
78595 
80615 
82546 
84394 
7 86167 
87871 
89510 
91089 
92613 


12 23524 
21405 
19385 
17454 
15606 
12 13833 
12129 
10490 
08911 
07387 


10 00001 
00001 
00001 
00001 
00001 


9 99999 
99999 
99999 
99999 
99999 


40 

39 
38 
37 
36 


1500 
1560 
1620 
1680 
1740 


7.86166 
87870 
89509 
91088 
92612 


12.13834 
12130 
10491 
08912 
07388 


5 31 442 
5.31442 
5 31 442 
5 31442 
5 31 441 
5 31 441 
5 31 441 
5.31441 
5 31441 
5 31 441 
5 31 441 
5 31441 
5 31 441 
5 31 441 
5 31441 


10 00001 
00001 
00001 
00001 
00002 
10 00002 
00002 
00002 
00002 
00002 
10 00002 
00002 
00003 
00003 
00003 
10 00003 
00003 
00003 
00003 
00004 
10.00004 
00004 
00004 
00004 
00004 
10.00005 
00005 
00005 
00005 
00005 


9 99999 
99999 
99999 
99999 
99998 


35 

34 
33 
32 
31 


1800 
I860 
1920 
1980 
2040 
2100 
2160 
2220 
2280 
2340 


30 

31 
32 
33 
34 


7 94084 
95508 
96887 
98223 
99520 


5 31 443 
5 31 443 
5.31443 
5.31443 
5 31 443 


12 059H 
04492 
03113 
01777 
00480 


7 94086 
95510 
96889 
98225 
99522 


12.05914 
04490 
03111 
01775 
00478 


9.99998 
99998 
99998 
99998 
99998 


30 

29 
28 
27 
26 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


8 00779 
02002 
03192 
04350 
05478 


5.31443 
5 31 443 
5 31443 
5 31 443 
5 31 443 
5 31443 
5 31 444 
5.31444 
5 31 444 
5 31 444 


11 99221 
97998 
96808 
95650 
94522 
11 93422 
92350 
91304 
90282 
89283 
11.88307 

8641< 
85505 
84609 


8.00781 
02004 
03194 
04353 
05481 
8 06581 
07653 
08700 
09722 
10720 


11 99219 
97996 
96cS06 
95647 
94519 
If 934 19 
92347 
91300 
90278 
89280 


9 99998 
99998 
99997 
99997 
99997 


25 

24 
23 
22 

21 


2400 
2460 
2520 
2580 
2640 


8 06578 
07650 
08696 
09718 
10717 


5 31 441 
5.31440 
5 31 440 
5.31440 
5.31440 
5 31440 
5 31 440 
5 31 440 
5.31440 
5 31440 
5 31 439 
5 31 439 
5.31439 
5.31439 
5 31 439 


9 99997 
99997 
99997 
99997 
99996 
9799996 
99996 
99996 
99996 
999% 
9 99995 
9999i 

9999! 
99995 


20 

19 
18 
17 
16 


2700 
2760 
2820 
2880 
2940 


45 

46 
47 
48 
49 


8 11693 
12647 
13581 
14495 
15391 


5.31444 
5 31444 
5.31444 
5 31444 
5.31444 


8 11696 
12651 
13585 
14500 
15395 


11 88304 
87349 
86415 
85500 
84605 
ET83727 
82867 
82024 
8119( 
80384 


15 

14 
13 
12 
11 


3000 
3060 
3120 
3180 
3240 


50 

51 
52 
53 
54 


8.16268 
17128 
17971 
18798 
19610 


5.31444 
5.31444 
5.31444 
5.31444 
5 31 444 


11.83732 
82872 
82029 
81202 
80390 


8.16273 
17133 
17976 
18804 
19616 


10 

9 

8 
7 
6 


3300 
3360 
3420 
3480 
3540 


55 

56 
57 
58 
59 


8.20407 
21189 
21958 
22713 
23456 


5,31444 
5.31444 
5 31 445 
5 31 445 
5 31 445 


11.79593 
78811 
78042 
77287 
76544 


8 20413 
21195 
21964 
22720 
23462 


5.31439 
5.31439 
5.31439 
5.31438 
5.31438 


11 79587 
78805 
78036 
77280 
76538 


10.00006 
00006 
00006 
00006 
00006 


9 99994 
99994 
99994 
99994 
99994 


5 

4 
3 
2 
1 


3600 


60 


24186 


5.31445 


75814 


24192 


5 31 438 


75808 


00007 


99993 







/ 


2cos 


/ sec 


I cot 


/ tan 


I CSC 


I Hill 


/ 



90 C 



89 C 



TABLE II 



178 



" 


' 


I sin 


log 8 


1 CSC 


ftan 


log T 


I cot 


I sec 


1 COS 


/ 


3600 
3660 
3720 
3780 
3840 




1 
2 
3 

4 


8.24186 
24903 
25609 
26304 
26988 


5.31445 
5.31445 
5.31445 
5 31 445 
5.31445 


11.75814 
75097 
74391 
73696 
73012 


8.24192 
24910 
25616 
26312 
26996 


5 31 438 
5.31438 
5.31438 
5 31 438 
5.31437 


11.75808 
75090 
74384 
73688 
73004 


0.00007 
00007 
00007 
00007 
00008 


9.99993 
99993 
99993 
99993 
99992 


60 

59 
58 
57 
56 


3900 
3960 
4020 
4080 
4140 


5 

6 
7 
8 
9 


8.27661 
28324 
28977 
29621 
30255 


5.31445 
5.31445 
5.31445 
5.31445 
5 31 445 


11.72339 
71676 
71023 
70379 
69745 
11.69121 
68505 
67897 
67298 
66708 


8.27669 
28332 
28986 
29629 
30263 


5.31437 
5.31437 
5.31437 
5.31437 
5.31437 


11.72331 
71668 
71014 
70371 
69737 


0.00008 
00008 
00008 
00008 
00009 


9.99992 
99992 
99992 
99992 
99991 


55 

54 
53 
52 
51 


4200 
4260 
4320 
4380 
4440 


10 

11 
12 
13 
14 


8.30879 
31495 
32103 
32702 
33292 


5 31 446 
5 31 446 
5 31 446 
5.31446 
5 31446 
5 31 446 
5 31446 
5.31446 
5 31446 
5.31446 


8.30888 
31505 
32112 
32711 
33302 


5.31437 
5.31436 
5.31436 
5 31 436 
5.31436 


11 69112 
68495 
67888 
67289 
66698 


10 00009 
00009 
00010 
00010 
00010 


9 99991 
99991 
99990 
99990 
99990 


50 

49 
48 
47 
46 


4500 
4560 
4620 
4680 
4740 


15 

16 
17 
18 
19 


8.33875 
34450 
35018 
35578 
36131 


11.66125 
65550 
64982 
64422 
63869 


8.33886 
34461 
35029 
35590 
36143 


5 31 436 
5.31435 
5.31435 
5 31 435 
5.31435 


11.66114 
65539 
64971 
64410 
63857 


10.00010 
00011 
00011 
00011 
00011 


9 99990 
99989 
99989 
99989 
99989 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 


4800 
4860 
4920 
4980 
5040 


20 

21 
22 
23 

24 


8.36678 
37217 
37750 
38276 
38796 


5 31 446 
5 31 447 
5 31 447 
5 31 447 
5 31 447 


11.63322 
62783 
62250 
61724 
61204 


8 36689 
37229 
37762 
38289 
38809 


5 31 435 
5.31434 
5 31 434 
5 31 434 
5 31 434 


11.63311 
62771 
62238 
61711 
61191 


10.00012 
00012 
00012 
00013 
00013 


9 99988 
99988 
99988 
99987 
99987 


5100 
5160 
5220 
5280 
5340 


25 

26 
27 
28 
29 


8 39310 
39818 
40320 
40816 
41307 


5 31 447 
5 31447 
5.31447 
5.31447 
5 31447 


11.60690 
60182 
59680 
59184 
58693 


8.39323 
39832 
40334 
40830 
41321 


5 31 434 
5 31433 
5 31 433 
5.31433 
5 31 433 


11.60677 
60168 
59666 
59170 
58679 


10 00013 
00014 
00014 
00014 
00015 


9.99987 
99986 
99986 
99986 
99985 


35 

34 
33 
32 
31 


5400 
5460 
5520 
5580 
5640 


30 

31 
32 
33 

34 


8 41792 
42272 
42746 
43216 
43680 


5.31447 
5.31448 
5 31 448 
5 31448 
5.31448 


11.58208 
57728 
57254 
56784 
56320 


8 41807 
42287 
42762 
43232 
43696 


5.3143S 
5 31 432 
5 31 432 
5 31 432 
5 31 432 


11 58193 
57713 
57238 
56768 
56304 


10 00015 
00015 
00016 
00016 
00016 


9.99985 
99985 
99984 
99984 
99984 


30 

29 
28 
27 
26 


5700 
5760 
5820 
5880 
5940 


35 

36 
37 
38 
39 


8 44139 
44594 
45044 
45489 
45930 


5 31 448 
5 31 448 
5 31 448 
5 31 448 
5 31 449 


11 55861 
55406 
54956 
54511 
54070 


8 44156 
44611 
45061 
45507 
45948 


5 31431 
5 31 431 
5 31 431 
5 31 431 
5 31 431 


11.55844 
55389 
54939 
54493 
54052 


10.00017 
00017 
00017 
00018 
00018 


9.99983 
99983 
99983 
99982 
99982 


25 

24 
23 
22 
21 


6000 
6060 
6120 
6180 
6240 


40 

41 
42 
43 
44 


8 46366 
46799 
47226 
47650 
48069 


5.31449 
5 31 449 
5.31449 
5.31449 
5 31 449 


11.53634 
53201 
52774 
52350 
51931 


8 46385 
46817 
47245 
47669 
48089 


5 31 430 
5 31 430 
5 31 430 
5.31430 
5 31429 


11 53615 
53183 
52755 
52331 
51911 


10 00018 
00019 
00019 
00019 
00020 


9.99982 
99981 
99981 
99981 
99980 


20 

19 
18 
17 
16 


6300 
6360 
6420 
6480 
6540 


45 

46 

47 
48 
49 


8 48485 
48896 
49304 
49708 
50108 


5.31449 
5.31449 
5.31450 
5 31450 
5.31450 


11.51515 
51104 
50696 
50292 
49892 


8 48505 
48917 
49325 
49729 
50130 


5.31429 
5.31429 
5 31 428 
5 31 428 
5.31428 


11 51495 
51083 
50675 
50271 
49870 


10 00020 
00021 
00021 
00021 
00022 


9 99980 
99979 
99979 
99979 
99978 


15 

14 
13 

12 
11 


6600 
6660 
6720 
6780 
6840 


50 

51 
52 
53 
54 


8.50504 
50897 
51287 
51673 
52055 


5 31450 
5 31450 
5.31450 
5.31450 
5 31 450 


11.49496 
49103 
48713 
48327 
47945 


8 50527 
50920 
51310 
51696 
52079 


5.31428 
5.31427 
5 31 427 
5 31 427 
5.31427 


11.49473 
49080 
48690 
48304 
47921 


10 00022 
00023 
00023 
00023 
00024 


9.99978 
99977 
99977 
99977 
99976 


10 

9 
8 
7 
6 


6900 
6960 
7020 
7080 
7140 


55 

56 
57 
58 
59 


8.52434 
52810 
53183 
53552 
53919 


5.31451 
5 31 451 
5.31451 
5 31451 
5.31451 


11.47566 
47190 
46817 
46448 
46081 


8.52459 
52835 
53208 
53578 
53945 


5.31426 
5.31426 
5.31426 
5 31 425 
5.31425 


11.47541 
47165 
46792 
46422 
46055 


10.00024 
00025 
00025 
00026 
00026 


9.99976 
99975 
99975 
99974 
99974 


5 

4 
3 
2 

1 


7200 


60 


54282 


5.31451 


45718 


54308 


5.31425 


45692 


00026 


99974 







r 


1 COS 


I sec 


/cot 


Ztan 


fcsc 


I sin 


/ 



91 c 



TABLE II 



177 C 



" 


' 


I sin 


log S 


I CSC 


Han 

8 54308 
54609 
55027 
55382 
55734 


logF 


Zcot 


I sec 


d 


1 COS 


' 


7200 
7260 
7320 
7380 
7440 




1 
2 
3 
4 
5 
6 
7 
8 
9 


8.54282 
54642 
54999 
55354 
55705 
8.56054 
56400 
56743 
57084 
57421 
8~57757 
58089 
58419 
58747 
59072 


5.31451 
5 31 451 
5 31452 
5 31 452 
5 31 452 
5.31 452 
5 31 452 
5 31 452 
5 31453 
5 31453 
5 31453 
5 31453 
5 31 453 
5 31453 
5 31 454 


11 45718 
45358 
45001 
44646 
44295 


5.31425 
5 31 425 
5.31424 
5 31 424 
5.31424 


11.45692 
45331 
44973 
44618 
442(56 


10 00026 
00027 
00027 
00028 
00028 
10 00029 
00029 
00030 
00030 
00031 
10 00031 
00032 
00032 
00033 
00033 


1 


1 




] 
u 
1 
u 

1 



1 




1 



1 
1 



1 



1 



1 



1 



1 



1 



1 
1 



1 



1 



1 

1 




1 



1 



1 
1 




9.99974 
99973 
99973 
99972 
99972 


60 

59 
58 
57 
56 


7500 
7560 
7620 
7680 
7740 


11 43946 
43600 
43257 
42916 
42579 


8 56083 
56429 
56773 
57114 
57452 


5 31 423 
5.31423 
5 31 423 
5 31 422 
5 31 422 


11 43917 
43571 
43227 
42886 
42548 
f i 42212 
41879 
41549 
41221 
40895 
U 740572 
40251 
39932 
39616 
39302 


J 99971 
99971 
99970 
99970 
99969 


55 

54 
53 
52 

51 


7800 
7860 
7920 
7980 
8040 


10 

11 
12 
13 
14 


11 42243 
41911 
41581 
41253 
40928 


8 57788 
58121 
58451 
58779 
59105 
8 59428 
50719 
60068 
60384 
60698 


5 31 422 
5.31421 
5.31421 
5.31421 
F> 31 421 


J 99969 
99968 
99968 
99967 
99967 


50 

49 
48 
47 
46 


8100 
8160 
8220 
8280 
8340 
8400 
8460 
8520 
8580 
8640 
8700 
87(50 
8820 
8880 
8940 


15 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


8.59895 
59715 
00033 
60349 
60662 
8 60973 
61282 
61589 
61S91 
62196 
8.62497 
62795 
63091 
63385 
63678 
8 63968 
64256 
64543 
61827 
65110 


5 31454 
5 31 454 
5 31 154 
5 31454 
5 31454 
5 31455 
5 31 455 
5 31455 
5 31455 
5 31455 
5 31 455 
5 31 456 
5 31456 
5 31456 
5 31 456 


11 40605 
40285 
39967 
39651 
39338 
11 39027 
38718 
38411 
38106 
37804 
11 37503 
37205 
36909 
36615 
36322 
11 36032 
35744 
35457 
35173 
31890 
1 1 34609 
31330 
34053 
33777 
33503 
If 33231 
32961 
32692 
32425 
32159 
11 31896 
31633 
31373 
31114 
30856 


5.31420 
5 31420 
5 31 420 
5 31419 
5 31419 


10 00033 
00034 
00034 
00035 
0003(5 


9 99967 
99966 
99966 
99965 
99964 


45 

44 
43 
42 
41 


8 61009 
61319 
61626 
61931 
62234 
8 62535 
62834 
63131 
63426 
63718 


5 31418 
5 31418 
5 31418 
5 31417 
5 314i7 
5 31417 
5 31416 
5 31416 
5 31416 
5 31415 
5 31415 
5 31415 
5 31414 
5.31 414 
5 31 413 
5 31 113 
5 31413 
5 31412 
5 31412 
5 31412 
5.31411 
5 31411 
5.31410 
5 31410 
5 31410 
5^31409 
5 31409 
5 31 408 
5 31 408 
5 31 408 
5 31 407 
5 31 407 
5 31406 
5 31 406 
5.31405 


11 38991 
38681 
38374 
38069 
37766 
11 37465 
37166 
36869 
36574 
36282 
11" 35991 
35702 
35415 
35130 
34846 
1T34565 
34285 
34007 
33731 
33457 


10 00036 
00037 
00037 
00038 
00038 
10 00039 
00039 
00040 
00040 
00041 


9 99964 
99963 
99963 
99962 
99962 
9" 99961 
99061 
99960 
99960 
99959 


40 

39 
38 
37 
36 
35 
34 
33 
32 
31 


9000 
9060 
9120 
9180 
9240 


5 31 456 
5 31456 
5 31457 
5 31 457 
5 31 157 
5 31457 
5 31457 
5 31458 
5 31458 
5 31458 
5 31 458 
5 31 158 
5 31 459 
5 31459 
5 31459 


8 64009 
(54298 
64585 
(54870 
65154 
8 65435 
65715 
65993 
66269 
66543 


10 00041 
00042 
00042 
00043 
00044 


9 99959 
99958 
99958 
99957 
99956 


30 

29 
28 
27 
26 


9300 
9360 
9420 
9480 
9540 


8 65391 
65670 
65947 
66223 
66497 
8 66769 
67039 
67308 
67575 
67841 
8 08 104 
68367 
68627 
68886 
69144 


10 00044 
00045 
00045 
00046 
00046 


9 99056 
99955 
99955 
99954 
99954 


25 

24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 


9600 
9660 
9720 
9780 
9810 
" 9900 
9960 
10020 
10080 
10140 
10200 
10260 
10320 
10380 
10440 


40 

41 
42 
43 
44 


8 66816 
67087 
67356 
67624 
67890 


11 33184 
32913 
32614 
32376 
32110 


10 00047 
00048 
00048 
00049 
00049 


9 99953 
99952 
99952 
99951 
99951 
9 99950 
99949 
99949 
99948 
99948 


45 

46 
47 
48 
49 
50 
51 
52 
53 
51 


5 31459 
5 31 459 
5.31460 
5 31460 
5.31460 


8 68154 
68417 
68678 
(58938 
69196 


11 31846 
31583 
31322 
31062 
30804 


10 00050 
00051 
00051 
00052 
00052 


8 69100 
69654 
69907 
70159 
70409 
8 70658 
70905 
71151 
71395 
71638 


5.31460 
5.31460 
5.31461 
5.31461 
5 31 461 
5731461 
5 31461 
5 31 462 
5 31 462 
5 31462 


11 30600 
30346 
30093 

29841 
29591 


8 69453 
69708 
69962 
70214 
70465 


11 30547 
30292 
30038 
29786 
29535 


10 00053 
00054 
00054 
00055 
00056 


9 99947 
99946 
99946 
99945 
99944 


10 

9 

8 
7 
6 


10500 
10560 
10620 
10680 
10740 


55 

56 
57 
58 
59 


11 29342 
29095 
28849 
28605 
28362 


8 70714 
70962 
71208 
71453 
71697 


5 31 405 
5 31 405 
5 31 404 
5 31404 
5 31 403 


11 29286 
29038 
28792 
28547 
28303 


10 00056 
00057 
00058 
00058 
00059 


9 99944 
99943 
99942 
99942 
99941 


5 

4 
3 
2 

1 

~T 


10800 


60 


71880 


5.31462 


28120 


71940 


5 31403 


28060 


00060 


99940 


' 


1 COS 




I sec 


1 cot 




I tan 


I CSC 


d 

1 


I sin 


' 



92 C 



87 C 



41 



3 



TABLK II 



176 C 



' 


I sin 
8. 


d 


I CSC 

11. 


I tan 
8. 


d 

r 


I cot 

11. 


I sec 
10. 


d 
1' 


I COS 

9. 


/ 




/> 


241 


i*rc 
239 


port 
237 


lona 
235 


iK 
234 


rts 
232 


229 




1 

2 
3 
4 
5 

6 
7 
8 
9 


71880 
72120 
359 
597 
834 


240 
239 
238 
237 
235 
234 
232 
232 
230 
229 
228 
220 
226 
224 
223 
222 
220 
220 
219 
217 
216 
216 
214 
213 
212 
211 
210 
209 
208 
208 
206 
20r> 
204 
203 
202 
201 
201 
199 
199 
197 
197 
1 1)6 
195 
194 
193 
192 
192 
190 
190 
189 
188 
187 
187 
186 
185 
184 
183 
183 
181 
181 


28120 
27880 
641 
403 
166 
26931 
697 
465 
233 
003 


71940 
72181 
420 
659 
896 


241 
239 
239 
237 
236 
234 
234 
232 
231 
229 
229 
227 
226 
225 
224 
222 
222 
220 
219 
219 
217 
216 
215 
214 
213 
211 
211 
210 
209 
208 
206 
206 
205 
204 
203 
202 
201 
201 
199 
198 
198 
196 
196 
195 
194 
193 
192 
192 
190 
190 
189 
188 
188 
186 
186 
185 
184 
184 
182 
182 


28060 
27819 
580 
341 
104 


00060 
060 
061 
062 
062 




1 


1 
1 



1 
1 



1 
1 



1 
1 
1 



1 
1 



1 
1 



1 
I 



1 
1 

1 


1 
1 
1 
1 


1 
1 

: 



1 
1 
1 
1 
1 



1 

1 
1 
1 
1 



1 
1 
1 

1 


99940 
940 
939 
938 
938 
"937 
936 
936 
935 
934 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 




I 

2 
3 
4 
5 

6 
7 
8 
9 

io 

11 

12 
13 
14 
15 

16 
17 
18 
19 


o 




















s 

12 
ll> 
"20 
24 
28 
32 
36 


8 
12 
10 


s 

12 
16 
20 
24 
28 
32 
36 


s 
12 
16 
20 
24 
27 
31 
35 


8 
12 
16 


8 
12 
15 


8 
11 
15 
19 
23 
27 
31 
34 


73069 
303 
535 
767 
997 


73132 
366 
600 
832 
74063 


26868 
634 
400 
168 
25937 


063 
064 
064 
065 
066 


20 
24 
28 
82 

36 


19 
23 
27 
31 
35 


19 
23 
27 
31 
35 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


74226 
454 
680 
906 
75130 


25774 
546 
320 
094 
24870 


292 
521 
748 
974 
75199 


708 
479 
252 
026 
24801 


066 
067 
068 
068 
069 
070 
071 
071 
072 
073 
074 
074 
075 
076 
077 


934 
933 
932 
932 
931 


50 

49 
48 
47 
46 


40 
44 
48 
52 
56 


40 
44 
48 
52 
56 


40 
43 
47 
51 
55 
59 
03 
67 
71 
75 
79 
83 
87 
91 
95 


39 
43 
47 
51 
55 
59 
63 
07 
70 
74 
78 
82 
86 
90 
94 


39 
43 
47 
51 
55 


39 
43 
46 
50 
54 


38 
42 
46 
50 
53 
57 
61 
65 
69 
73 


353 
575 
795 
76015 
234 


647 
425 
205 
23985 
766 


423 
645 
867 
76087 
306 


577 
355 
133 
23913 
694 
" 475 
258 
042 
22827 
613 


930 
929 
929 
928 
927 
~926 
926 
925 
924 
923 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 


(50 
64 
68 
72 
70 


GO 
64 
68 
72 
70 


59 
62 
00 
70 
74 
"78 
82 
86 
90 
94 


58 
62 
66 
70 
73 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


451 

667 
883 
77097 
310 


549 
333 
117 
22903 
690 


525 

742 
958 
77173 
387 


20 

21 
22 
23 
24 


80 
84 
88 
92 
90 
100 
104 
108 
112 
116 


80 
84 
88 
92 
96 


77 
81 
85 
89 
93 


76 
80 
84 
88 
92 
95 
99 
103 
107 
111 
114 
118 
122 
126 
130 
134 
137 
141 
145 
149 


522 
733 
943 
78152 
360 
78568 
774 
979 
79 183 
386 


478 
267 
057 
21848 
640 
21432 
226 
021 
20817 
614 


600 
811 
78022 
232 
441 
"78649 
855 
79061 
266 
470 


400 
189 
21978 
768 
559 
~2f351 
145 
20939 
734 
530 


077 
078 
079 
080 
080 


923 
922 
921 
920 
920 


35 

34 
33 
32 
31 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


100 
104 
108 
112 
116 
120 

123 

127 
131 
135 
139 
143 
147 
151 
155 


99 
103 
107 
111 
115 
118 
122 
126 
130 
134 


98 
102 
106 
110 
114 
7l8 
121 
125 
129 
133 


97 
101 
105 
109 
113 


97 
101 
104 
108 
112 
116 
120 
124 
128 
131 


00081 
082 
083 
083 
084 


99919 
918 
917 
917 
916 


30 

29 
28 
27 
26 


120 
125 
129 
133 
137 


117 
121 
125 
129 
133 
137 
140 
144 
148 
152 


35 

36 
37 
38 
39 


588 
789 
990 
80189 
388 


412 
211 
010 
19811 
612 


673 
875 
80076 
277 
476 


327 
125 
19924 
723 
524 


085 
086 
087 
087 
088 


915 
914 
913 
913 
912 


25 

24 
23 
22 
21 


141 
145 
149 
153 
157 
161 
165 
169 
173 
177 
181 
185 
189 
193 
197 
201 
205 
209 
213 
217 


138 
142 
146 
150 
154 
158 
162 
166 
170 
174 
178 
182 
186 
190 
194 


137 
141 
145 
149 
153 


135 
139 
143 
147 
151 
155 
159 
162 
166 
170 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


585 
782 
978 
81173 
367 


415 
218 
022 
18827 
633 


674 
872 
81068 
264 
459 


326 
128 
18932 
736 
541 


089 
090 
091 
091 
092 


911 
910 
909 
909 
908 


20 

19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


159 
163 
167 
171 
175 
179 
183 
187 
191 
195 


157 
161 
164 
168 
172 


156 
160 
164 
168 
172 
175 
179 
183 
187 
191 
"195 
199 
203 
207 
211 


153 
156 
160 
164 
168 


560 
752 
944 
82134 
324 


440 
248 
056 
17866 
676 


653 
846 
82038 
230 
420 


347 
154 
17962 
770 
580 


093 
094 
095 
096 
096 


907 
906 
905 
904 
904 


176 
180 
184 
188 
192 


174 
178 
182 
186 
189 


172 
176 
179 
183 
187 
191 
195 
198 
202 
206 


50 

51 
52 
53 
54 


513 
701 
888 
83075 
261 


487 
299 
112 
16925 
739 


610 
799 
987 
83175 
361 


390 
201 
013 
16825 
639 


097 
098 
099 
100 
101 


903 
902 
901 
900 
899 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


199 
203 
207 
211 
215 
219 
223 
227 
231 
235 
239 


198 
201 
205 
209 
213 


196 
200 
204 
208 
212 
215 
219 
223 
227 
231 


193 
197 
201 
205 
209 


55 

56 
57 

58 
59 


446 
630 
813 
996 
84177 


554 
370 
187 
004 
15823 


547 
732 
916 
84100 
282 


453 
268 
084 
15900 

718 


102 
102 
103 
104 
105 


898 
898 
897 
896 
895 


5 

4 

I 
1 


221 
225 
229 
233 
237 


217 
221 
225 
229 
233 


215 
218 
222 
226 
230 


213 

217 
220 

224 
228 


210 
214 
218 
221 
225 


60 


84358 


15642 


84464 


15536 


00106 


99894 







60 


241 


237 


235 


234 


232 


229 


/ 


8. 

1 COS 


> 


11. 

/sec 


8. 
I cot 


d 

r 


11. 

ftan 


10. 

I CHC 


d 


9. 

I sin 


/ 




// 


241 


239 

Pr< 


237 

)por 


235 

tiom 


234 

ilPi 


232 

irts 


229 



93 C 



86 C 



42 



TABLE II 



// 


227 


225 


223 


220 


217 


215 


213 


211 


208 


3 


>rtio 
203 


nal 
201 


l>art 
199 


s 
197 


195 


193 


192 


189 


187 


185 


1831181 


~<r 

i 




4 



4 





















3 



3 



3 



3 



3 



3 



3 



3 



3 



3 



3 



3 



3 



3 


2 
3 


8 
11 


8 
11 


11 


11 


11 


11 


11 


11 


10 


10 


10 


10 


10 


10 


C 
10 


6 
10 


6 
10 


C 
9 


6 
9 


6 
9 



9 


C 
9 


4 


15 


15 


15 


15 


14 


14 


14 


14 


14 


14 


14 


13 


13 


13 


13 


13 


13 


13 


12 


12 


12 


12 


5 


19 


19 


19 


18 


18 


18 


18 


18 


17 


17 


17 


17 


17 


16 


16 


16 


16 


16 


16 


15 


15 


15 


6 


23 


22 


22 


22 


22 


22 


21 


21 


21 


21 


20 


20 


20 


20 


20 


19 


19 


19 


19 


18 


18 


18 


7 


26 


26 


26 


26 


25 


25 


25 


25 


24 


24 


24 


23 


23 


23 


23 


23 


22 


22 


22 


22 


21 


21 


8 


30 


30 


30 


29 


29 


29 


28 


28 


28 


27 


27 


27 


27 


26 


26 


26 


26 


25 


25 


25 


24 


24 


9 


34 


34 


33 


33 


_ 33 


32 


32 


32 


31 


31 


30 


30 


30 


30 


29 


29 


29 


28 


28 


28 27 


27 


10 


"38 


38 


~37 


37 


36 


36 


36 


35 


35 


34 


34 


34 


33 


33 


32 


~32 


32 


32 


31 


31 f 30 


3~0 


11 


42 


41 


41 


40 


40 


39 


39 


39 


38 


38 


37 


37 


36 


36 


36 


35 


35 


35 


34 


34! 34 


33 


12 


45 


45 


45 


44 


43 


43 


43 


42 


42 


41 


41 


40 


40 


39 


39 


39 


38 


38 


37 


37 37 


36 


13 


49 


49 


48 


48 


47 


47 


46 


46 


45 


45 


44 


44 


43 


43 


42 


42 


42j 41 


41 


40 40 


39 


14 


53 


52 


52 


51 


51 


50 


50 


49 


49 


48 


^7 


47 


46 


46 46 


45 


45 44 44 


43 43 


42 


15 


"57 


56 


5ft 


"55 


54 


" 54 


53 


~53 


52 


51 


51 


50 


50 


49 


49 


48 


48 " 47 


47 


46 


46 


~45 


16 


61 


60 


59 


59 


58 


57 


57 


56 


55 


55 


54 


54 


53 


53 


52 


51 


51 


50 


50 


49 49 


48 


17 


64 


64 


63 


62 


61 


61 


60 


(iO 


59 


58 


58 


57 


56 


56 


55 


oi 


54 54 


53 


52 52 51 


18 


68 


68 


67 


66 


65 


64 


64 


63 


62 


62 61 


60 


60 


59 


581 58 


58; 57 


56 


50 55 54 


19 


72 


71 


71 


70 


69 


68 


67 


67 66 


65 64 


64 


63 


62 


621 61 


61 ! 00 


59 


59 58, 57 


20 


~76 


75 


"74 


73 


~72 


~72 


71 


" 70 >9 


69 


68 


07 i 66 


66 65 i 64 


04 7i.3 


02 


021 01 ' 60 


21 


79 


79 


78 


77 


76 


75 


75 


74 73 


72 


71 


70 70 


69 


68 


68 


07, 66 


65 


65 i 64 63 


22 


83 


82 


82 


81 


80 


79 


78 


77 70 


76 74 


74 


73 


72 


72 


71 


70: 691 09 


681 67 i 66 


23 


87 


86 


85 


84 


83 


82 


82 


81 80 


79 


78 


77 


76 


76 


75 


74 


74 72 


72 


71 ! 70| 69 


24 


91 


90 


89 


88 


87 


86 


85 


84 


83 


82 


81 


80 


80 


79 


78 


77 


77 70 j 75 


74 i 73 72 


25 


95 


94 


93 


92 


~90 


~9;) 


~89 


88 


87 


~XO 


~85 


~84 


83 


82 


81 


8() 


8(1 79 


78 


77 


76 75 


26 


98 


98 


97 


95 


94 


93 


92 


91 


90 


89 i SS 


87 


80 


85 


84 84 


83 82 


81 


80 


79 


78 


27 


102 


101 


100 


99 


98 


97 


96 


95 


94 


93 


91 


90 


90 


89 


88 87 


80 85 


84 


83 


82 


81 


28 


106 


105 


104 


103 


101 


100 


99 


98 


97 


96 


95 


94 


93 


92 


91 


90 


90; 88 


87 


86 


85 


84 


29 


110 


109 


108 


106 


105 


104 


103 


102 


101 


100 


98 


97 


96 


95 


94 


93 


93; 91 


90 


89 


88 


87 


30 


114 


112 


112 


110 


~108 


108 


10(5 


106 


104 


103 


102 


100 


100 


98 


98 


90 


90 94 


94 


~92 


~1)2 


90 


31 


117 


116 


115 


114 


112 


111 


110 


109 


107 


106 


105 


104 


103 


102 101 


100 


99 98 


97 


96 


95 


94 


32 


121 120 


119 


117 


116 


115 


114 


113 


111 


110 


108 


107] 100 


105 104 


103 


102; 101 


100 


99! 98 


91 


33 


12,5 124 


123 


121 


119 


118 


117 


116 


114 


113 


112 


111! 109 


108 107 


106 


lOGj 104 


103 


102i 101 


100 


34 


129 


128 


126 


125 


123 


122 


121 


120 


US 


117 


115 


1141113 


112 110 


109 


109i 107 


106 


105! 104 


103 


35 


132 


"131 


130 


128 


127 


125 


124 


123 


"121 


120 iTs 


117 


116 


115 


114! 113 


112" no 


109 


108~107 


106 


36 


130 


135 


134 


132 


130 


129 


128 


127 


125 


124 122 


121 


119 


118 


H7illO 


115 113 


112 


lllj 110| 109 


37 


140 


139 


138 


136 


134 


133 


131 


130 


12S 


127 125 


124 


123 


121 


120 119 


118 117 


115 


114' 113| H2 


38 


144 


142 


141 


139 


137 


130 


135 


134 


132 


130 


129 


127 


126 


125 


124 


122 


122 120 


; 118 


117' 116' 115 


39 


148 


146 


145 


143 


141 


140 


13S 


137' 135 


134 


132 


131 


129 


128 


127 


125 


125 123 


: 122 


120 119J 118 


40 


151 


~150 


119 


147 


145 


143 


142 


T41 ~13<) 


737 


135 


134 


133 


131 


130 


129 


~128 126 


i 125 


723 122| 121 


41 


155 


154 


152 


150 


148 


147 


146 


144, 142 


HI 


139 


137 


136 


135 


133 


132 


131 129! 128 


126 125 124 


42 


159 


158 


156 


154 


152 


150 


149 


148 


140 


144 


142 


141 


139 


138 


136 


135 


134 132 


1 131 


130| 128 127 


43 


103 


161 


160 


158 


156 


154 


153 


151 


149 


148 


145 


144 


143 


141 


140 


138 


138, 135 


134 


133 


131 


130 


44 


166 


165 


164 


161 


159 


158 


156 


155 


153 


151 


149 


147 


14C 


144 


143 


142 


141J 139 


, 137 


136 


134 133 


45 


170 


169 


167 


165 


163 


161 


160 


15S 


156 


155 


152 


151 


149 


148 


146 


145 


144 


142 


~140 


~139 


137 


136 


46 


174 


172 


171 


169 


166 


165 


163 


162 


159 


158 


156 


154 


153 


151 


150 


148 


147 


145 143 


142 


140 


139 


47 


178 


176 


175 


172 


170 


168 


167 


165 


163 


161 


159 


157 


156 


154 


153 


151 


150 


148 146 


145 


143 


142 


48 


182 


180 


178 


176 


174 


172 


170 


169 


166 


165 


162 


161 


159 


158 


156 


154 


loi 


151 


150 


148 


146 


145 


49 


185 


184 


182 


180 


177 


176 


174 


172 


170 


16S 


166 


164 


163 


161 


159 


158 


157 


154 


; 153 


151 


149 


148 


50 


~189 


188 


186 


183 


181 


179 


178 


176 


173 


172 


Ttt9 


168 


ToT) 


164 


162 ~161 


100 158! 156 


154 


152 


151 


51 


193 


191 


190 


187 


184 


183 


181 


179 


177 


175 


173 


171 


169 


167 


166! 164 


163; 161 


'159 


157 


156 


154 


52 


197 


195 


193 


191 


188 


186 


185 


183 


180 


179 


17b 


174 


172 


171 


169 i 167 


166 


164; 162 


160 


159 


157 


53 


201 


199 


197 


194 


192 


190 


188 


186 


184 


182 


179 


178 


176 


174 


1721 170 


170 


167i 165 


163 


162 


160 


54 


204 


202 


201 


198 


195 


194 


192 


190 


187 


185 


183 


181 


179 


177 


176J 174 


173 


170, 168 


166 


165 


163 


55 


208 


206 


204 


202 


199 


197 


195 


193 


191 


189 


186 


~184 


182 


181 


179 


177 


176 


173! 171 


170 


168 


166 


56 


212 


210 


208 


205 


203 


201 


199 


197 


194 


192 


189 


188 


18f 


184 


182 


180 


179 


176 i 175 


173 


171 


169 


57 


216 


214 


212 


209 


206 


204 


202 


200 


198 


196 


193 


191 


189 


187 


185 


183 


182 


180 


178 


176 


174 


172 


58 


219 


218 


216 


213 


210 


208 


206 


204 


201 


199 


196 


194 


192 


190 


188 


187 


186 


183 


181 


179 


177 


175 


59 


223 


221 


219 


216 


213 


211 


209 


207 


205 


203 


200 


198 


19b 


194 


192 


190 


189 


186 


184 


182 


180 


178 


60 


227 


225 


223 


220 


217 


215 


213 


211 


208 


206 


203 


201 


199 


197 


195 


19: 


192 


18fl 


187 


185 


183 


181 


// 


227 


225 


223J 


220 


217 


215 


213 


211 


208 


206 


203 


201 


199 


197 


195 


193 


192 


188 


187 


185 


183 


181 






>roportional Parts 





43 



TABLE II 



175 C 



' 


I sin 
8. 


d 


/ CSC 

11. 


Han 
8. 


d 


Jcot 

11. 


/ sec 
10. 


d 

1' 


/ cos 
9. 


60 

59 

58 
57 
56 




" 


182 


Prc 
181 


>por1 
179 


lona 
177 


il Pa 
176 


rts 
175 


174 




1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


84358 
539 
718 
897 
85075 
252 
429 
605 
780 
955 


181 
179 
179 
178 
177 
177 
176 
175 
175 
173 
173 
173 
171 
171 
171 
169 
109 
109 
107 
168 
166 
166 
165 
164 
164 
163 
163 
162 
162 
160 
161 
159 
139 
159 
158 
157 
157 
156 
155 
155 
155 
154 
153 
153 
152 
152 
151 
151 
150 
150 
149 
149 
148 
147 
147 
147 
14(j 
146 
145 
145 


15642 
461 
282 
103 
14925 


84464 
646 
826 
85006 

185 


182 
180 
180 
179 
178 
177 
177 
176 
176 
174 
174 
174 
172 
172 
171 
171 
170 
169 
169 
168 
107 
167 
166 
165 
165 
165 
163 
103 
103 
161 
102 
ICO 
100 
160 
159 
158 
158 
157 
157 
156 
155 
155 
155 
153 
154 
153 
152 
152 
151 
151 
150 
150 
149 
148 
149 
147 
147 
147 
146 
146 


15536 
354 

174 
14994 
815 


00106 
107 
108 
109 
109 


1 
1 
1 



1 
1 

1 
1 

1 
1 
1 
1 
1 



i 

i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
l 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 
i 

2 

1 
1 
1 

1 
1 
1 
1 
1 
1 
1 
1 
1 
2 


99894 
893 
892 
891 
891 




1 

2 
3 

4 



3 
6 
9 

12 
15 
IS 
21 
24 
27 
~30 
33 
30 
39 
42 



3 
6 
9 
12 



3 
(J 
9 
12 



3 
G 
9 
12 
15 
18 
21 
24 
27 



3 
6 
9 
12 



3 
6 
9 
12 
15 
18 
20 
23 
20 



3 

9 
12 


748 
571 
395 
220 
045 


363 
540 
717 
893 
86069 


637 
460 
283 
107 
13931 


110 

111 
112 
113 
114 


890 
889 
888 
887 
880 


55 

54 
53 
52 
51 
50 
49 
18 
47 
10 


5 

6 
7 
8 
9 


15 
18 
21 

24 
27 
~30 
33 
30 
39 
42 


15 
18 
21 
24 

27 
30 
33 
30 
39 
42 
45 
48 
51 
54 
57 
00 
03 
00 
09 
72 
"75 
78 
81 
81 
87 
90 
92 
95 
98 
101 
104 
107 
110 
113 
110 


15 

18 
21 
23 
20 
29 
32 
35 
38 
41 


14 
17 
20 
23 
20 


86128 
301 
474 
645 
816 


13872 
699 
526 
355 

184 


243 
417 
591 
763 
935 


757 
583 
409 
237 
065 


115 
116 
117 
118 
119 
"""120 
121 
121 
122 
123 
124 
125 
126 
127 
128 


885 
884 
883 
882 
881 


10 

11 
12 
13 
14 
15 
10 
17 
18 
19 
20 
21 
22 
23 
24 
25 
20 
27 
28 
29 
30 
31 
32 
33 
34 
35 
30 
37 
38 
39 
40 
41 
42 
43 
44 
45 
40 
47 
48 
49 
50 
51 
52 
53 
54 


30 
32 
35 
38 
41 


29 
32 
35 
38 
41 
44 
47 
50 
52 
55 
"58 
01 
04 
07 
70 


29 
32 
35 
38 
41 
41 
40 
49 
52 
55 
58 
01 
(>4 
07 
70 
72 
75 
78 
81 
84 


987 
87156 
325 
494 
661 
829 
995 
88161 
326 
490 


013 
12844 
675 
506 
339 


87106 
277 
447 
616 

785 


12894 
723 
553 
384 
215 


880 
879 
879 

878 
877 


45 

44 
43 
12 
41 
40 
39 
JS 
37 
3H 


45 
49 
52 
55 

58 
01 
04 
07 
70 
73 


45 
48 
51 
54 
57 
GO 
03 
lit) 
09 
72 
75 
7S 
si 

81 

87 


44 
47 
50 
53 
50 


44 
47 
50 
53 
50 
59 
02 
05 
07 
70 
73 
70 
79 
82 
85 


171 
005 
11839 
674 
510 


953 
88120 
287 
453 

618 


047 
11880 
713 
547 
382 


870 

875 
874 
873 
872 


59 
02 
05 
08 
71 


25 

26 
27 

28 
29 


654 
817 
980 
89142 
304 


346 
183 
020 

10858 
696 


783 
948 
89111 
274 
437 


217 
052 
10889 
726 
563 


129 
130 
131 
132 
133 


871 
870 
800 
808 
867 


35 

34 
33 
32 
31 


7(5 
79 
82 
85 

ss 


74 
77 
80 
S3 
SO 

"ss 

91 
91 
97 
100 
103 
100 
109 
112 
115 


73 
70 
79 

82 
85 

"ss 

90 
93 
90 
99 
102 
105 
108 
111 
114 

117 
120 
122 


30 

31 
32 
33 
34 


89464 
625 
784 
943 
90102 


10536 
375 
216 
057 

09898 


89598 
760 
920 
90080 
240 


10402 
240 
080 
09920 
760 


00134 
135 
136 
137 

138 


99800 
865 
804 
863 
802 


30 

29 
28 
27 
20 

25 

24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 





i 
1 


91 
94 
97 
100 
103 


90 
91 
97 
100 
103 


88 
91 
94 
97 
100 


87 
90 
93 
90 
99 


35 

36 
37 
38 
39 
10 
41 
42 
43 
44 
15 
46 
47 
48 
49 
50 
51 
52 
53 
54 


260 
417 
574 
730 

885 


740 
583 
426 
270 
115 


399 
557 
715 
872 
91029 


601 
443 
285 
128 
08971 


139 
140 
141 
142 
143 


801 
800 
859 

858 
857 


100 
109 
112 
115 
US 
121 
124 
127 
130 
133 


100 
109 
112 
113 
IIS 


103 
100 
109 
111 
114 
117 
120 
123 


102 
KM 
107 
110 
113 
110 
119 
122 


91040 
195 
349 
502 
655 


08960 
805 
651 
498 
345 


185 
340 
495 
650 
803 


815 
660 
505 
350 
197 


144 
145 
146 
147 

148 


850 

855 
854 
853 

852 


121 
124 
127 
130 
133 
130 
139 
142 
145 
148 
151 
151 
157 
100 
103 


119 
122 
125 


118 
121 
124 


131 
134 
137 
140 
113 
140 
149 
152 
155 
158 
101 
104 
107 
170 
173 
176 
179 


130 


129 
132 
135 
138 
111 
141 
147 
150 
153 
155 
158 


128 
131 
134 
137 
140 
143 
140 
149 
152 
155 
158 
IfiT) 
103 
100 
109 
172 


128 
130 
133 
130 
139 
142 


807 
959 
92110 
261 
411 


193 
041 
07890 
739 
589 


957 
92110 
262 
414 
565 


043 
07890 
738 
586 
435 


149 
150 
152 
163 
154 


851 
850 

848 
847 
846 


137 
140 
143 
MO 
149 
152 
155 
158 
1G1 
104 
107 
170 
173 
170 
179 
182 


133 
130 
139 
112 
145 
148 
150 
153 
150 
159 


561 
710 
859 
93007 
154 


439 
290 
141 
06993 
846 


716 
866 
93016 
165 
313 


284 
134 
06984 
835 
687 


155 
156 
157 
158 
159 


845 
844 
843 
842 
841 


145 
148 
151 
154 
157 


55 

56 
57 
58 
50 


301 
448 
594 
740 

885 


699 
552 
406 
260 
115 


462 
609 
756 
903 
94049 


538 
391 
244 
097 
05951 


160 
161 
162 
163 
164 


840 
839 

838 
837 
836 


55 

50 
57 
58 
59 
60 


100 
109 
172 

175 
178 


102 
1(15 
108 
171 
174 


101 
104 
107 
170 
173 


100 
102 
165 
168 
171 
^174 
174 


60 

/ 


94030 


05970 


94195 


05805 


00166 


99834 




/ 




181 


177 


170 


175 


8.. 
/cos 


d 


11. 

Zsec 


8. 
Zcot 


d 

1' 


11. 

Ztan 


10. 

/ CSC 


d 
1' 


9. 

1 sin 


// 


182 


181 

Pr< 


179 

)por 


177 

tiona 


176 

il Pa 


175 

irts 



94 C 



85 C 



44 



TABLE II 





Proportional Parts 




173 


172 


171 


169 


167 


166 


165 


1631162 


160 


159 


158 


157 


55 


153 


162 


61 


150 


149 


47 


1461 


145 





" 6 





"" 



























































1 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


2 


2 


2 


2 


2 


2 























5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


3 


9 


9 


9 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


7 


7 


7 


7 


4 


12 


11 


11 


11 


11 


11 


11 


11 


11 


11 


^11 


11 


10 


10 


10 


10 


10 


10 


10 


_10 


Hi 


10 


5 


14 


14 


14 


14 


^14 


14 


14 


"~14 


14 


13 


~13 


13 


13 


~13 


13 


13 


13 


12 


12 


12 


~Ti 


12 





17 


17 


17 


17 


17 


17 


10 


10 


1C 


10 


16 


10 


16 


16 


15 


15 


15 


15 


15 


15 


15 


14 


7 


20 


20 


20 


20 


19 


19 


19 


19 


19 


19 


19 


18 


18 


18 


18 


18 


18 


18 


17 


17 


17 


17 


8 


23 


23 


23 


23 


22 


22 


22 


22 


22 


21 


21 


21 


21 


21 


20 


20 


20 


20 


20 


20 


19 


19 


9 


2G 


20 


20 


25 


25 


25 


25 


24 


24 


24 


24 


24 


24 


L3 


23 


23 


23 


22 


22 


22 


22 


22 


10 


~29 


" 29 


28 


28 


28 


~2S 


28 


' 27 


"~27 


"27 


~2J 


20 


20 


20 


26 


25 


25 


25 


25 


~24 


" 24 


24 


11 


32 


32 


31 


31 


31 


30 


30 


30 


30 


29 


29 


29 


29 


28 


28 


28 


28 


28 


27 


27 


27 


27 


12 


35 


34 


34 


34 


33 


33 


33 


33 


32 


32 


32 


32 


31 


31 


31 


30 


30 


30 


30 


29 


29 


29 


13 


37 


37 


37 


37 


30 


30 


30 


35 


35 


35 


34 


34 


34 


34 


33 


33 


33 


32 


32 


32 


32 


31 


14 


40 


40 


iO 


39 


39 


39 


38 


38 


38 


37 


3" 


37 


JJ7 


36 


_36 


35 


35 


35 


35 


34 


34 


34 


15 


43 


43 


"43 


42 


"42 


42 


~41 


41 


40 


40 


40 


40 


39 


39 


38 


3S 


38 


~38 


37 


37 


30 


36 


10 


46 


40 


40 


45 


45 


44 


44 


43 


43 


43 


42 


42 


42 


41 


41 


41 


40 


40 


40 


39 


39 


39 


17 


49 


49 


4S 


48 


47 


47 


47 


4G 


4G 


45 


45 


45 


44 


44 


43 


43 


43 


42 


42 


42 


41 


41 


IS 


52 


52 


51 


51 


50 


50 


50 


4\\ 


49 


48 


48 


47 


47 


46 


40 


40 


45 


45 


45 


44 


44 


44 


10 


55 


54 


54 


54 


53 


53 


52 


52 


51 


51 


50 


50 


_ 50 


49 


48 


48 


48 


48 


47 


47 


46 


46 


20 


5S 


~57 


57 


~5? 


'50 


55 


"55 


54 


51 


53 


53 


53 


~52 


52 


~51 


51 


50 


50 


50 


49 


"49 


48 


21 


(il 


00 


00 


59 


58 


5,X 


58 


57 


57 


56 


50 


55 


55 


54 


54 


53 


53 


52 


52 


51 


51 


51 


22 


03 


03 


03 


1)2 


01 


01 


GO 


GO 


51) 


59 


58 


58 


58 


57 


56 


56 


55 


55 


55 


54 


54 


53 


23 


06 


00 


on 


05 


04 


M 


03 


02 


02 


01 


61 


01 


GO 


59 


59 


58 


58 


58 


57 


56 


56 


56 


24 


09 


69 


08 


Ii8 


07 


00 


GO 


05 


G5 


G4 


64 


03 


63 


02 


. G1 


61 


60 


00 


GO 


59 


58 


_58 


25 


72 


~~72 


71 


70 


70 


" 09 


G9 


08 


GS 


07 


~66 


66 


65 


"65 


"64 


63 


63 


62 


62 


61 


61 


60 


20 


75 


75 


74 


73 


72 


72 


72 


71 


70 


69 


69 


68 


68 


67 


GO 


60 


65 


65 


65 


04 


63 


03 


27 


7S 


77 


77 


70 


75 


75 


74 


73 


73 


72 


72 


71 


71 


70 


69 


08 


68 


08 


67 


6( 


66 


65 


28 


si 


SO 


SO 


79 


7S 


77 


77 


7G 


7G 


75 


74 


74 


73 


72 


71 


71 


70 


70 


70 


63 


68 


68 


29 


S4 


S3 


S3 


82 


81 


SO 


80 


79 


78 


77 


77 


70 


76 


75 


74 


73 


73 


72 


72 


71 


71 


7G 


30 


so 


80 


"Sfi 


st 


84 


S3 


82 ~S2 


" 81 


80 


~80 


"79 


"78 


78 


76 


"70 


70 


75 


74 


~7t 


~73 


72 


31 


S9 


S9 


88 


87 


80 


80 


85 


8 1 84 


83 


82 


82 


81 


80 


79 


79 


78 


78 


77 


76 75 


75 


32 


92 


92 


in 


90 


so 


89 


88 


S7 SG 


85 


So 


84 


84 


83 


82 


81 


81 


80 


79 


78 


78 


77 


33 


93 


95 


91 


93 


92 


91 


91 


90 8H 


88 


87 


87 


8G 


85 


84 


84 


83 


82 


82 


81 


80 


80 


34 


98 


97 


97 


9li 


95 


){ 


91 


92 


92 


91 


90 


9( 


89 


88 


87 


86 


8P 


85 


84 


83 


83 


82 


35 


101 


100 


!(;(. 


99 


97 


97 


91 > 


~9.> 


94 


93 


93 


92 


92 


90 


89 


"89 


~88 


88 


87 


86 


" 85 


85 


36 


KM 


103 


103 


101 


100 


100 


99 


98 


97 


96 


9o 


95 


94 


93 


92 


91 


91 


9( 


89 


88 


88 


87 


37 


107 


100 


105 


10-1 


103 


102 


102 


101 


100 


99 


98 


97 


97 


9( 


94 


94 


93 


92 


92 


91 


90 


89 


38 


110 


109 ! 10S 


107 


10G 


105 


104 


103 


103 


101 


101 


KM 


9< 


98 


97 


96 


9b 


95 


94 


93 


92 


92 


39 


112 


112 


111 


110 


109 


10S 


107 


100 


105 


104 


103 


103 


102 


101 


99 


99 


98 


98 


97 


96 


95 


94 


40 


115 


115 


114 


113 


7TT 


111 


110 


109 


108 


107 


106 


105 


105 


103 


102 


101 


101 


lol 


99 


98 


97 


~97 


41 


us 


lisl 117 


115 


114 


113 


113 


111 


111 


109 


109 


108 


107 


m 


105 


104 


103 


10L 


102 


100 


100 


99 


42 


121 


120 


120 


118 


117 


110 


110 


111 113 


112 


111 


111 


110 


108 


107 


10( 


lOb 


105 


104 


103 


102 


102 


43 


124 


123 


123 


121 


120 


119 


118 


117 


IK 


115 


114 


113 


113 


111 


110 


109 


108 


108 


107 


105 


105 


104 


44 


127 


120 


125 


124 


122 


122 


121 


120 


119 


117 


117 


llf 


115 


114 


112 111 


111 


110 


109 


108 


107 


106 


45 


130 


129 


12S 


127 


T25 


12-1 


124 


122 12L 


120 


TlT) 


Tfs 


Tis 


7To|ll5Tl4 


113 


112 


112 


110 


110 


109 


40 


133 


132! 131 


130 


128 


127 


120 125 


124 


123 


122 


121 


120 


119 


117 


117 


lie 


115 


114 


113 


112 


111 


47 


13fi 


135 


134 


132 


131 


13( 


129 128 


127 


125 


125 


124 


123 


121 


120 


119 


118 


118 


117 


115 


114 


114 


48 


138 


138 


137 


135 


134 


133 


132 


130 


13C 


128 


127 


120 


12( 


124 


122 


122 


121 


120 


119 


118 


117 


11G 


49 


141 


140 


140 


138 


130 


13d 


135 


133 


131 


131 


130 


129 


128 


127 


125 


124 


12.' 


122 


122 


120 


119 


llf 


50 


"144 


143 


142 


141 


139 


13S 


138 


130 


135 


133 


132 


132 


131 


129 


128 


127 


120 


125 


124 


122 


122 


121 


51 


147 


140 


145 


144 


142 


141 


140 


139 


13S 


136 


135 


134 


133 


132 


130 


129 


128 


128 


12" 


125 


124 


123 


52 


150 


149 


148 


111) 


145 


144 


143 


141 


14( 


139 


138 


137 


13( 


134 


133 


132 


131 


130 


129 


127 


127 


126 


53 


153 


152 


151 


149 


148 


147 


140 


141 


141 


141 


14( 


140 


139 


137 


135 


134 


13: 


132 


13L 


13C 


129 


128 


54 


150 


155 


154 


152 


150 


149 


148 


147 


14G 


144 


14J 


142 


141 


140 


138 


13 


136 


135 


134 


132 


131 


130 


55 


159 


158 


'l57 


155 


153 


152 


151~ 


119 


148 


147 


146 


145 


144 


142 


1~4(1 


139 


Isa 


138 


137 


135 


134 


133 


56 


101 


101 


100 


158 


150 


155 


154 


152 


151 


149 


148 


147 


147 


145 


143 


14 


14 


14C 


139 


137 


136 


135 


57 


104 


103 


102 


101 


159 


158 


157 


155 


154 


152 


151 


150 


149 


147 


145 


144 


143 


142 


142 


140 


139 


138 


58 


107 


100 


105 


103 


101 


100 


100 


158 


157 


155 


154 


153 


152 


150 


148 


14 


146 


14S 


144 


142 


141 


140 


59 


170 


109 


108 


100 


104 


103 


102 


100 


159 


157 


15f 


155 


I 54 


152 


150 


149 


148 


14* 


147 


145 


144 


143 


60~ 


173 


172 


"m 


109 


107 


IOC 


1G5 


103 


TGL 


16t 


159 


~158 


157 


155 


153 


15 


15 


75C 


149 


147 


146 


145 


ft 


173 


172 


171 


109 


167 


166 


165 


163 


m 


160 


159 


158 


167 


155 


153 


152 


151 


15C 


149 


14? 


146 


146 




Proportional Parts 



45 



TABLE II 



174 



' 


I sin 
8. 


d 
r 


I CSC 

11. 


/tan 
8. 


d 

l' 


/cot 
11. 


/sec 
10. 


d 

1' 


I COS 

9. 


/ 




" 


145 


ir< 

144 


>por 
143 

" 
2 
5 


tiom 
142 


2 
5 


UPa 
141 


LTtS 

140 


139 


( 


94030 
174 
317 
461 
603 


144 
143 
144 
142 
143 
141 
142 
141 
140 
140 
139 
139 
139 
138 
138 
137 
137 
136 
136 
136 
135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 
131 
130 
130 
I5W 


05970 
826 
683 
539 
397 


94195 
340 
485 
630 
773 


145 
145 
145 
143 
144 
143 
142 
142 
142 
141 
140 
141 
139 
140 
138 
139 
138 
137 
138 
136 
137 
135 
136 
135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 
131 
130 
130 
130 
129 
128 
129 
128 
127 
128 
127 
126 
126 
126 
126 
Ift5 


05805 
660 
515 
370 
227 


00166 
167 
168 
169 
170 


1 
1 
1 
1 
1 
1 
1 
2 
1 
1 
1 
1 
1 
1 
2 
1 
1 
1 
1 
1 
2 
1 
1 

2 
1 

1 
1 

2 

2 
1 

2 
1 

2 

1 
1 
1 
2 
1 
1 
1 
2 
1 
1 
2 
1 
1 
2 
1 
1 
2 

d" 

r 


99834 
833 
832 
831 
830 


CO 

59 
58 
57 
56 






1 

2 
3 

4 



2 
5 



2 
5 



5 



2 
5 



2 
5 


t 


10 
"12 
14 
17 
19 
22 
"24 
27 
29 
31 
34 
30 
39 
41 
44 
40 
18 
51 
53 
50 
58 
0. 
(W 
03 
US 
70 
72 
75 
77 
80 
82 
85 
87 
89 
92 
94 
97 
99 
102 
104 
100 
109 
111 
111 
110 
118 
121 
123 
120 
128 
130 


10 


10 


9 


9 


9 
12 
14 
10 
19 
21 
23 
26 
28 
30 
33 
35 
37 
40 
42 
44 


9 


5 

6 

r 

8 
9 


746 
887 
95029 
170 
310 


254 
113 
04971 
830 
690 


917 
95060 
202 
344 

486 


083 
04940 
798 
656 
514 


171 
172 
173 
175 
176 


829 
828 
827 
825 
824 


55 

54 
53 
52 
51 




5 

6 

7 
8 
9 


12 
14 
17 
19 
22 
24 
2(5 
29 
31 
34 
30 
38 
41 
43 
40 


12 
14 
17 
19 
21 
24 
26 
29 
31 
33 


12 
14 
17 
19 
21 
24 
20 
28 
31 
33 
30 
38 
40 
43 
45 
47 
50 
52 
54 
57 
59 
02 
04 
00 
0! 
71 
73 
70 
78 
80 
83 
85 
88 
90 
92 
95 
97 
99 
102 
104 
100 
109 
111 
114 
110 
118 
121 
123 
125 
128 


12 
14 
10 
19 
21 


12 
14 
10 
19 
21 


H 

1 f 
t 


450 
589 
728 
867 
96005 


550 
411 
272 
133 
03995 


627 
767 
908 
96047 

187 


373 
233 
092 
03953 
813 


177 
178 
179 
180 
181 


823 
822 
821 
820 
819 


50 

49 
48 
47 
46 




10 

11 
12 
13 
14 


24 
20 
28 
31 
33 
35 
38 

s 

45 


23 
25 
28 
30 
32 


15 

16 
17 
13 
19 


143 
280 
417 
553 
689 


857 
720 
583 
447 
311 


325 
464 
602 
739 

877 


675 
536 
398 
261 
123 


183 
184 
185 
186 
187 


817 
816 
815 
814 
813 


45 

44 
43 
42 
41 


15 

16 
17 
18 
19 


30 
3S 
41 
43 
45 
48 
50 
52 
55 
57 
(>() 
l>2 
14 
l>7 
09 
72 
74 
70 
79 
81 
83 
80 
88 
91 
93 
95 
98 
100 
102 
105 
107 
110 
112 
114 
117 
119 
122 
124 
120 
129 


35 
37 
39 
42 
44 


20 

21 
22 
23 
24 


825 
' 960 
97095 
229 
363 


175 
040 
02905 
771 
637 


97013 
150 
285 
421 
556 


02987 
850 
715 
579 
444 


188 
190 
191 
192 
193 


812 
810 
809 
808 
807 


40 

39 
38 
37 
3(i 


20 

21 
22 
23 

24 


48 
50 
53 
55 
58 


47 
49 
52 
54 
50 
59 
01 
)3 
1)0 
08 
70 
73 
75 
78 
80 
82 
85 
87 
89 
92 
94 
90 
99 
101 
103 
100 
108 
110 
113 
115 
118 
120 
122 
125 
127 


47 
49 
51 
54 
SB 
58 
01 
03 
05 
08 
70 
72 
75 
77 
79 
82 
84 
80 
89 
91 
93 
90 
98 
100 
103 
105 
107 
110 
112 
111 
117 
119 
121 
124 
126 


40 
49 
51 
53 
50 


25 

26 
27 
28 
2f 


496 
629 
762 
894 
98026 


504 
371 
238 
10(5 
01974 


691 
825 
959 
98092 
225 


309 
175 
041 
01908 
775 


194 
196 
197 
198 
199 


806 
804 
803 
802 
801 


35 

34 
33 
32 
31 


25 

20 
27 
28 
29 


o;; 

02 
05 
07 
70 
72 
74 
77 
79 
82 
84 
80 
89 
91 
94 


58 
60 
03 
05 
07 


30 

31 
32 
33 
34 


98157 
288 
419 
549 
679 


01843 
712 
581 
451 
321 


98358 
490 
622 
753 

884 


01642 
510 
378 
247 
116 


00200 
202 
203 
204 
205 
""207 
208 
209 
210 
212 


99800 
798 
797 
796 
795 
~"793 
792 
791 
790 
788 


30 

21) 
2S 
27 
26 


30 

31 
32 
33 

34 


70 
72 
74 
70 
79 


35 

36 
37 
38 
39 


808 
937 
99066 
194 
322 


129 
129 
128 
128 
1V>8 


192 
063 
00934 
806 

678 


99015 
145 
275 
405 
534 


00985 
855 
725 
595 
466 


25 

24 
23 
22 
21 


35 

36 
37 
38 
39 


81 
83 
80 
88 
90 


40 

41 
42 
43 
44 


450 
577 

704 
830 
956 


127 
127 
126 
126 
126 
125 
125 
124 
125 
123 
124 
123 
23 
122 
M, 


550 
423 
296 
170 
044 


662 
791 
919 


338 
209 
081 


213 
214 
215 
217 
218 


787 
786 
785 
783 
782 


20 

19 
18 
17 
16 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


'.Hi 
98 
101 
103 
100 
108 
110 
113 
115 
1'18 
120 
122 
125 
127 
130 


93 
95 
97 
100 
102 
104 
107 
109 
111 
114 


00046 
174 


99954 
826 


45 

46 
47 

48 
49 


00082 
207 
332 
456 
581 
704 
828 
951 
01074 
196 


99918 
793 
668 
544 
419 
296 
172 
049 
98926 
804 


301 
427 
553 
679 
805 


699 
573 
447 
321 
195 
070 
98945 
821 
697 
573 


219 
220 
222 
223 
224 


781 
780 
778 
777 
776 
775 
773 
772 
771 
769 


15 

14 
13 
12 
11 
10 
9 
8 
7 
6 


50 

51 
52 
53 
54 


930 
01055 

179 
303 
427 


125 
124 
124 
124 
123 
123 
123 
22 
122 
22 

T 

r 


225 
227 
228 
229 
231 


110 
118 
120 
123 
125 


55 

56 
57 
58 
59 


318 
440 
561 
682 
803 


122 
121 
121 
121 
20 

~d~ 

r 


682 
560 
439 
318 
197 


550 
673 
796 
918 
02040 


450 
327 
204 
082 
97960 


232 
233 
235 
236 
237 


768 
767 
766 
764 
763 


5 

4 
3 
2 
1 


55 

56 
57 
58 
59 


133 
135 
138 
140 
143 
"145 
145 


132 
134 
137 
139 
142 
"144 
144 
Pr< 


131 
133 
130 
138 
141 


130 
133 
135 
137 
140 
"142 


129 
132 
134 
130 
139 


128 
131 
133 
135 
138 
140 
140 
rts 


127 
130 
132 
134 
137 
139 
139 


60 


01923 


98077 


02162 
~9. 
/cot 


97838 
10. 
Man 


00239 
10. 

/ CMC 


99761 





60 


143 
143 
>por 


141 
141 
ilPa 


9. 

/ cos 


10. 

I sec 


9. 

/ sin 


142 

tioni 



95 C 



84' 



46 



TABLE II 





Proportional Parts 


// 


138 


137 


136 


135 


134 


133 


132 


131 


130 


129 


128 


127 


126 


125] 


124 


123 


122 


121 


120 


2 


1 


~T 

































































i 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 


2 








3 


5 

7 


7 


5 

7 


7 


7 


7 


7 


7 


6 


6 


6 


6 


6 


6 


6 


6 


6 


6 


6 










4 


9 


9 


9 


9 


9 


9 


_9 


9 


9 


9 


9 


8 


8 


8 


8 


8 


8 


8 


8 








5 


12 


11 


ii" 


11 


" 11 


11 


11 


11 


11 


ii 


11 


11 


~lb 


"16 


~~10 


~7o 


10 


10 


"10 





~0 


6 


14 


14 


14 


14 


13 


13 


13 


13 


13 


13 


13 


13 


13 


12 


12 


12 


12 


12 


12 








7 


16 


16 


16 


16 


16 


10 


15 


15 


15 


15 


15 


15 


15 


15 


14 


14 14 


14 


14 








8 


18 


18 


18 


18 


18 


18 


18 


17 


17 


17 


17 


17 


17 


17 


17 


10! 10 


16 


10 








9 


21 


21 


20 


20 


20 


20 


20 


20 


20 


19 


19 


19 


19 


19 


19 


18 


18 


18 


18 








10 


23 


~~23 


23 


"22 


22 


22 


22 


22 


22 


22 


2*1 


21 


21 


21 


21 


20 


20 


20 


"20 


~0 





11 


25 


25 


25 


25 


25 


24 


24 


24 


24 


24 


23 


23 


23 


23 


23 


23 


22 


22 


22 








12 


28 


27 


27 


27 


27 


27 


20 


20 


20 


26 


a- 


25 


25 


25 


25 


25 


24 


24 


24 








13 


30 


30 


29 


29 


29 


29 


29 


28 


28 


28 


2S 


28 


27 


27 


27 


27 


26 


26 


26 








14 


32 


32 


32 


32 


31 


31 


31 


31 


30 


30 


30 


30 


29 


29 


29 


29 


28 


28 


28 








15 


~34 


~~34 


"34 


34 


34 


33 


"33 


~33 


32 




32 


"^2 


32 


31 


31 


31 


30 


30 


30 


^r 


~0~ 


10 


37 


37 


36 


36 


30 


35 


35 


35 


35 


34 


34 


34 


34 


33 


33 


33 


33 


32 


32 


i ! o 


17 


39 


31) 


39 


38 


38 


38 


37 


37 


37 


37 


36 


36 


36 


35 


35 


35 


35 


34 


34 


1 





18 


41 


41 


41 


40 


40 


40 


40 


39 


39 


39 


38 


38 


38 


38 


37 


37 


37 


36 


30 


1 





11) 


44 


43 


43 


43 


42 


42 


42 


41 


41 


41 


41 


40 


40 


40 


39 


39 


39 


38 


38 


1 





20 


40 


~4G 


"45 


45 


"45 


44 


44 


44 


43 


43 


43 


~42 


42 


42 


~41 


41 


41 


40 


40 


~r 


T> 


21 


48 


48 


48 


471 47 


47 


40 


46 


46 


45 


45 


44 


44 


44 


43 


43 


43 


42 


42 


1 





22 


51 


50 


50 


50 


49 


49 


48 


18 


48 


47 


47 


47 


40 


46 


45 


45 


45 


44 


44 


1 





23 


53 


53 


52 


52 


51 


51 


51 


50 


50 


49 


49 


49 


48 


48 


48 


47 


47 


46 


46 


1 





24 


55 


55 


,54 


54 


54 


53 


53 


52 


52 


52 


51 


51 


50 


50 


50 


49 


49 


48 


48 


1 





25 


58 


57 


"57 


56 


50 


55 


55 


55 


~54 


54 


53 


" 53 


52 


52 


52 


51 


51 


~50 


50 


1 





26 


oo 


59 


59 


58 


58 


58 


57 


57 


50 


56 


55 


55 


55 


54 


54 


53 


53 


52 


52 


1 





27 


02 


02 


01 


01 


60 


GO 


59 


59 


5H 


58 


58 


57 


57 


50 


50 


55 


55 


54 


54 


1 





28 


ot 


04 1 63 


03 


03 


62 


02 


01 


01 


60 


00 


59 


59 


58 


58 


57 


57 


56 


56 


1 





29 


07 


66 


66 


05 


05 


04 


64 


63 


03 


02 


62 


01 


61 


00 


60 


59 


59 


5S 


58 


1 





30 


~~G9 


68 


08 


08 


07 


~ 00 


00 


Of 


05 


64 


"ol 


~64 


~63 


~02 


~02 


~02 


01 


GO 


00 


1 


o" 


31 


71 


71 


70 


70 


69 


09 


68 


68 


07 


07 


00 


66 


05 


05 


04 


64l G3 1 63 


02 


1 


1 


32 


74 


73 


73 


72 


71 


71 


70 


70 


09 


69 


08 


OS 


07 


07 


GO 


60 


05 05 


64 


1 




33 


76 


75 


75 


74 


74 


73 


73 72 


72 


71 


70 


70 


69 


09 


08 


68 


67 


07 


06 


1 


1 


34 


78 


78 


77 


70 


70 75 


75, 74 


74 


73 73 


72 


71 


71 


70 


70 


09 


69 


68 


1 


1 


35 


80 


80 


70 


" 79 


"78 


78 


77i~~76 


70 


75 75 


74 


74 


73 72 


72 


~71 ~71 


70 


1 


1 


36 


83 


82 


82 


81 80 


8(! 


79! 79 


78 


77 77 


70 


70 75' 74 


74 


73! 73 


72 


1 


1 


37 


85 


84 


84 


83 83 


82 


81 


81 


SO 


SO 


79 78 


78 77 7f 


76 


75 75 


74 


1 


1 


38 


87 


87 


80 


SO 


85 


84 


84 


S3 


82 


82 


81! 80 


SO 79 79 


78 


77 


77 


76 1 


1 


39 


90 


89 


88 


88 


87 


80 


86 


85 


84 


84 


83 


83 


82 


81 


81 


80 


79 


79 


78 1 1 


1 


40 


92 


91 


"91 


90 


89 


"~89 


"88 


87 


~S7 


80 


85 


~85 


84 


83 


83 


82 ~8l 


"81 


s~6 i " 


"T 


41 


94 


94 


03 


92 


92 


91 


90 


90 


89 


88 


87 


87 


80 


85 Sf 


84 83 83 


82' 1 


1 


42 


97 


96 


95 


94 


94 


93 


92 


92 


91 


90 


90 


89 


88 


88 1 87 


86j 85 85 


84 1 


1 


43 


99 


98 


97 


07 


90 


95 


95 


91 


93 


92 


92 


91 


90 


90 


8f 


88 87 


87 


SOJ 1 


I 


44 


101 


100 


100 


99 


98 


98 


97 


90 


95 


95 


94 


93 


92 


92 


JL 1 


90 


89 


81 


88 


1 


1 


45 


~104 


103 


"102 


~iol 


100 


100 


~99 


"98 


98 


97 


90 


95 


94 


" 94 


9; 


~ 92 


92 


91 


90 


2 


1 


46 


10(5 


105 


KM 


104 


103 


102 


101 


100 


100 


99 


98 


97 


97 


90 


95 


94 


94 


93 


92 


2 


1 


47 


108 


107 


107 


100 


105 


104 


103 


103 


102 


101 


100 


99 


99 


98 


97 


96 


90 


95 


94 


2 


1 


48 


110 


110 


109 


108 


107 


106 


106 


105 


104 


103 


102 


102 


101 


100 


91 


98 


98 


97 


96 


2 


1 


49 


113 


112 


111 


110 


109 


109 


108 


107 


106 


105 


105 


104 


103 


102 


101 


100 


100 


99 


98 


2 


1 


50 


115 


114 


113 


112 


112 


111 


110 


109 


108 


108 


107 


100 


105 


104 


103 


102 


102 


101 


100 


2 


1 


51 


117 


116 


116 


115 


114 


113 


112 


111 


110 


110 


109 


108 


107 


100 


105 


lOo 


104 


103 


102 


2 


1 


52 


120 


119 


118 


117 


110 


115 


114 


114 


113 


112 


111 


110 


109 


108 


10" 


107 


106 


105 


104 


2 


1 


53 


122 


121 


120 


119 


118 


117 


117 


116 


115 


114 


113 


112 


111 


110 


IK 


109 


108 


107 


106 


2 


1 


54 


124 


123 


122 


122 


121 


120 


119 


118 


117 


116 


115 


114 


113 


112 


11L 


111 


110 


109 


108 


2 


1 


55 


126 


126 


125 


T24 


~123 


~12~2 


"l21 


120 


7l9 


118 


117 


116 


116 


~115 


114 


113 


112 


111 


110 


2 


1 


56 


129 


128 


127 


126 


125 


124 


123 


122 


121 


120 


119 


119 


118 


117 


11G 


115 


114 


113 


112 


2 


1 


57 


131 


130 


129 


128 


127 


120 


125 


124 


124 


123 


122 


121 


120 


119 


118 


117 


116 


115 


114 


2 


1 


58 


133 


132 


131 


130 


130 


129 


128 


127 


126 


125 


124 


123 


122 


121 


12( 


111 


118 


117 


116 


2 


1 


59 


136 


135 


134 


133 


132 


131 


130 


129 


128 


127 


126 


125 


124 


123 


12L 


121 


120 


IK 


118 


2 


1 


60 


"138 


137 


136 


135 


134 


"133 


"132 


'131 


130 


129 


128 


"127 


~126 


125 


124 


123 


"122 


121 


120 


2" 


1 


" 


138 


137 


136 


135 


134 


133 


132 


131 


130 


129 


128 


127 


126 


125 


124 


123 


122 


121 


120 


2 


~1 




Proportional Parts 



47 



TABLE II 



173 C 



-I r |, d . 


I csc 
10. 


/tan 
9. 


d 
1' 


/cot 
10. 


I sec 
10. 


d 
1 


/cos 
9. 
99761 
760 
759 
757 
756 


f 




1 

2 
3 
4 
5 


7 
8 
9 


121 


2 

4 


8 


Propo 
120 

" " o 

4 
6 

8 


rtiona 
119 

"6" 

2 

4 

8 
10 
12 
14 
10 
18 
20 
22 
21 
20 
2S 
30 
32 
34 
30 
3S 
10 
12 
U 
-10 

is 

30 
52 
34 
50 
38 
(>() 
01 
03 
1,5 

h7 

09 
71 
73 
75 
77 
7!) 
81 
S3 
85 
87 
85) 
91 
5)3 
95 
97 
99 
101 
103 
105 
107 
109 
111 
113 
115 
117 
119 
119 
tional 


1 Part 
118 

"""o 

2 
4 
6 

8 


s 
117 

d 

2 
4 



8 


i 
a 

4 


\ 01923 
1 02043 
163 
283 
402 


'120 



119 
118 
119 
118 
117 
118 
117 
117 
116 
116 
116 
116 
115 
115 
114 
115 
113 
114 
114 
113 
112 
113 
112 
112 
112 
111 
111 
111 
110 
110 
110 
110 
109 
109 
109 
108 
109 
108 
107 
108 
107 
107 
100 
107 
100 
105 
106 
105 
105 
105 
105 
104 
104 
104 
103 
103 
103 


98077 
97957 
837 
717 
598 


02162 
283 
404 
525 
645 


121 
121 
121 
120 
121 
119 
120 
119 
118 
119 
18 
118 
117 
18 
16 
17 
10 
110 
10 
15 
lf> 
llo 

114 
14 
113 
114 
113 
12 
113 
12 
112 
112 
111 
11 
111 
10 
111 
10 
109 
110 
09 
109 
108 
09 
08 
108 
07 
108 
07 
106 
107 
06 
106 
106 
106 
105 
105 
105 
104 

T 
1' 


97838 
717 
596 
475 
355 


00239 
240 
241 
243 
244 


1 
1 

2 

1 
1 
2 
1 
1 
2 

1 

1 
2 
I 
1 

1 
1 
2 

1 

1 

1 

1 
2 
1 
2 

1 
2 
1 
1 

1 

2 
1 

1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 
2 
1 

1 
2 
1 
2 
1 
2 


60 

59 
58 
57 
50 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


520 
639 

757 
874 
992 


480 
361 
243 
126 
008 
96891 
774 
658 
542 
426 


766 
885 
03005 
124 
242 
361 
479 
597 
714 
832 


234 
115 
96995 
876 
758 
639 
521 
403 
286 
168 
052 
95935 
819 
703 
587 


245 
247 
248 
249 
251 
252 
253 
255 
256 
258 


755 
753 
752 
751 
749 


55 

54 
53 
>2 


to 

14 
10 
IS 
20 
22 
24 
20 
2S 
30 
32 

34 
30 

38 
40 
42 
44 
40 
4S 

50 
52 
54 
50 
58 
00 ~ 

03 

05 
07 
09 

71~ 
73 
75 
77 
7!) 
SI " 
S3 
83 
87 
Si) 

5)1 
93 
95 
97 
99 


10 
12 
14 
10 
18 
20 
22 
21 
20 

as 

30 
32 
34 
30 
38 
40 
42 
44 
40 
48 

50 
52 
54 
50 

58 
00 
02 
01 

or. 

OS 
70 
72 
74 

70 
78 
SO 

8-1 

SS 
DO 
02 
5)4 
00 
i)S 
100 
i02 
104 
100 
108 

no" 

112 
114 
110 
118 
120 

120 

*ropor 


10 
12 
14 
10 

18 


10 
12 
14 
10 

18 


03109 
226 
342 
458 
574 


748 
747 
745 
744 
742 


?0 

49 

48 

1-7 
10 


10 

11 
12 
13 
14 
1.5 
10 
17 
18 
19 


20 
22 
24 
20 
28 
29 
31 
33 
35 
37 
39" 
It 
43 
45 
47 


20 
21 
23 
25 
27 
29 
31 
33 
35 
37 
" 3!) 
41 
43 
45 
47 
4!) 

53 
55 
57 
58 
00 
02 
04 
00 
08~ 
70 
72 
74 
70 
78 
SO 
82 
84 
80 
88" 
5)0 
92 
5)4 
5)0 


690 
805 
920 
04034 

149 


310 
195 
080 
95966 
851 


948 
04065 
181 
297 
413 


259 
260 
262 
263 
264 


741 
740 
738 
737 
736 


4o 

I'l 
43 
12 
11 


20 

21 

22 
23 
24 


262 
376 
490 
603 
715 


738 
624 
510 
397 
285 


528 
643 
758 
873 
987 


472 
357 
242 
127 
013 


266 
267 
269 

270 
272 


734 
733 
731 
730 

72S 


40 

39 
38 
37 
30 
35 
34 

10 

32 
31 
30 
20 
28 
27 
20 


20 

21 
22 
23 
24 


5 

26 
27 
28 
29 
30 
31 
32 
33 
34 


828 
940 
05052 
164 
275 
"05386 
497 
607 
717 
827 


172 
060 
94948 
836 
725 
94614 
503 
393 
283 
173 


05101 
214 
328 
441 
553 
05666 
778 
890 
06002 
113 


94899 
786 
672 
559 
447 
94334 
222 
110 
93998 
887 


273 

274 
276 
277 
279 
002SO 
282 
283 
284 
286 


727 
726 
724 
723 
721 
99720 
718 
717 
716 
714 


25 

20 
27 
2S 
29 
30 
31 
32 
33 
34 
33 
30 
37 
38 
39 
40 
41 
42 
43 
44 
45 
40 
47 
48 
49 
50 
51 
52 
r>3 
f>4 
55 
50 
57 
.58 
59 
60 


49 

I 

59 

;;i 
". 

09 
71 

s 

77 
75) 
81 
S3 
85 
87 
85) 
90 
92 
94 
<)0 
5)8 
100 
102 
104 
100 
108 
110 
112 
114 
110 
118 
US 
Parts 


35 

36 
37 
38 
39 


937 
06046 
155 
264 
372 


063 
93954 
845 
736 
628 


224 
335 
445 
55ii 
666 


776 
665 
555 
444 
334 
225 
115 
006 
92897 
789 
680 
572 
464 
357 
249 


2871 
289 
290 
292 
203 
295 
290 
298 
299 
301 
302 
304 
305 
307 
308 
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311 
313 
. 314 
316 
317 
319 
320 
322 
323 
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713 
711 
710 
70S 
707 
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704 
702 
701 
090 
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096 
695 
693 
692 
090 
089 
087 
686 
084 


5 

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23 
22 
21 
10 
19 
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17 
10 
15 
14 
13 
12 
11 

10 

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8 
7 



40 

2 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


481 
589 
696 
804 
911 
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124 
231 
337 
442 


519 
411 
304 
196 
089 
92982 
876 
769 
663 
558 


775 
885 
994 
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211 
320 
428 
536 
643 
751 


548 
653 
758 
863 
968 


452 
347 
242 
137 
032 


858 
964 
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177 
283 


142 
036 
91929 
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717 


101 
103 

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107 
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111 
113 
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117 
110 
121 
121 
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101 
103 
105 
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111 
113 
115 


55 

56 
57 
58 
59 
60 


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176 
280 
383 
486 


91928 
824 
720 
617 
514 


389 
495 
600 
705 
810 


611 
505 
400 
295 
190 


083 
681 
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117 



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" 


116 


115 


114 


113 


112 


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111 


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110 


nalP 
109 


arts 
108 


107 


106 


105 


104 


2 


1 




1 

2 
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2 



2 



2 



2 



2 



2 



2 



2 



2 



2 



2 



2 



2 
3 
5 
















6 

8 


6 

8 


6 

8 


6 
8 


6 


6 


5 


5 


5 


5 


5 


5 


5 

6 
7 
8 
9 


10 
12 
14 
15 
17 


10 
12 
13 
15 

17 


9 
11 
13 
15 
17 


9 
11 
13 
16 
17 


9 
11 
13 
15 
17 


9 
11 
13 
15 
17 

""is' 

20 
22 
24 
26 
28 
30 
31 
33 
35 


9 
11 
13 
15 
17 


9 
11 
13 
15 
16 


9 

11 
13 
14 
16 


9 
11 
12 
14 
16 


9 
11 
12 
14 
16 


9 
10 
12 
14 
18 


9 
10 
12 
14 
16 
















10 

11 
12 
13 
14 


19 
21 
23 
25 
27 
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31 
33 
35 
37 


19 
21 
23 
25 
27 
"29 
31 
33 
34 
36 


19 
21 
23 
25 

27 
29 
30 
32 
34 
36 


19 
21 
23 
24 
26 


19 
21 
22 
24 
26 


18 
20 
22 
24 
26 


IS 
20 
22 
24 
_2* 
27 
29 
31 
33 
35 
36 
38 
40 
42 
44 


18 
20 
22 
23 
25 


18 
20 
21 
23 
25 


18 
19 
21 
23 
25 


18 
19 
21 
23 
24 


17 
19 
21 
23 
24 








""6 

1 
1 

1 
1 









15 

16 
17 
18 
19 


28 
30 
32 
34 
30 


28 
30 
32 
34 
35 
37 
39 
41 
43 
45 


27 
29 
31 
33 
35 
"37" 
39 
40 
42 
44 


27 
29 
31 
32 
34 
"38 
38 
40 
41 
43 
"45 
47 
49 
50 
52 


27 
29 
30 
32 
34 


27 
28 
30 
32 
34 


26 
28 
30 
32 
33 


26 
28 
29 
31 
33 
35" 
36 
38 
40 
42 








"o" 







20 

21 
22 
23 
24 


39 
41 
43 
44 

40 
" 48" 
50 
52 
54 
50 
58 
00 
02 

r>4 

00 
OK 
70 
72 
73 
75 


38 
40 
42 
44 
46 

~4~iT 

50 
52 
54 
50 


38 
40 
4li 
44 
46 


38 
40 
41 
43 
45 
~47 
49 
51 
53 
55 


37 
39 

41 
43 
44 


36 
37 
39 
41 
43 


35 
37 
39 
41 
42 


35 

37 
38 
40 
42 


1 
1 
1 
1 
1 
"l 
1 
1 
1 
1 


25 

20 
27 
28 
29 


47 
49 
51 
53 
55 
~ 57 
59 
61 
63 
65 


47 
49 
50 
52 
54 


46 
48 
50 
52 
54 


46 
48 
49 
51 
53 
~ 55 
57 
59 
61 
62 


45 
47 
49 
51 
53 


45 

46 
48 
50 
52 
54 
55 
57 
59 
61 
62 
64 
66 
08 
70 


44 
46 
48 
49 
51 


44 
46 
, 47 
49 
51 


43 
45 
47 
49 
50 









30 

31 
32 
33 
34 
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30 
37 
38 
39 


58 
59 
01 
03 
65 


50 
58 
60 
62 
64 


56 
58 
60 
02 
03 


56 
57 
59 
61 
63 


54 
50 
58 
60 
62 
" 64 
65 
67 
69 
71 


54 
56 
58 
59 
61 


53 
55 
57 

58 
60 


52 
54 
56 
58 
60 


52 
54 
55 
57 
59 


1 

1 
1 
1 
1 




1 

1 
1 
1 


67 
69 
71 
73 
75 


67 
68 
70 
72 
74 


66 
68 
70 
72 
73 


65 
07 
69 
71 
73 
75 
77 
78 
80 
82 


65 
67 
08 
70 
72 


64 
66 
68 
70 
72 


(>3 
65 
67 
68 
70 


62 
64 
65 
67 
69 


61 
63 
65 
66 

68 


61 
62 
64 
66 

68 


1 
1 

1 
1 
1 

1 

1 
1 
1 


1 
1 
1 
1 
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40 

41 
42 
43 
44 
45 
40 
47 
48 
49 


77 
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81 
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85 
87" 
89, 
91 
93 
95 


77 
79 
80 
82 
81 
'80 
88 
90 
92 
94 


76 
78 
80 
82 
84 
85 
87 
89 
91 
93 


75 
77 

79 
81 
83 


74 

76 
78 
80 
81 
83 
85 
87 
89 
91 


73 
75 
77 

79 
81 
83 
84 
86 
88 
90 


73 

74 
76 
78 
80 
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84 
85 
87 
89 


72 
74 
76 
77 
79 


71 
73 
75 

77 
78 


71 
72 
74 
76 

78 


70 
72 
7-i 
75 

77 


69 
71 
73 
75 
76 


1 
1 
1 

1 
1 


85 
87 
89 
90 
92 


84 
86 
88 
90 
91 


81 
83 
85 
86 

88 


80 
82 
84 
80 
87 


79 
81 
83 

85 
87 


79 
80 
82 
84 
86 


78 
80 
81 
83 
85 


2 
2 
2 
2 
2 


1 
1 
1 

1 
1 


50 

51 
52 
53 
54 
55 
50 
57 
58 
59 
60 


97 
99 
101 
102 
104 

"Too" 

108 
110 
112 
114 
110 


90 
98 
100 
102 
104 


95 
97 
99 
101 
103 


94 
96 
98 
100 
102 
104 
105 
107 
109 
111 
113 


93 
95 
97 
99 
101 


92 
94 
96 
98 
100 
102 
104 
105 
107 
109 
TlT 


92 
93 
95 
97 
99 

Toi 

103 
105 
106 
108 


91 
93 
94 
96 
98 
100 
102 
104 
105 
107 
109 


90 
92 
94 
95 
97 
99 
101 
103 
104 
106 


89 
91 
93 
95 
96 


88 
90 
92 
94 
95 


88 
89 
91 
93 
94 


87 
88 
90 
92 
94 


2 
2 
2 
2 
2 


1 
1 
1 
1 
1 


105 
107 
109 
111 
113 


105 
106 
108 
110 
112 

in 


103 
105 
106 
108 
110 


98 
100 
102 
103 
105 


97 
99 
101 
102 
104 


90 
98 
100 
102 
103 
105 


95 
97 
99 
101 
102 
104~ 

To! 


2 
2 
2 

2 
2 


1 
1 
1 
1 
1 
i 


115 


112 


110 


108 


107 


106 


2 


" 


116 


115 


114 


113 


112 


111 
P 


110 

roporti 


109 

onal 1 


108 

'arts 


107 


106 


105 


2 


1 



49 



TABLE II 



172 C 



/ 

1 

1 

2 
3 

4 


/sin 
9. 
08589 
692 
795 
897 
999 


d 

103 
103 
102 
102 
102 
101 
102 
101 
101 
100 
101 
100 
100 
99 
100 

99 

99 
98 
90 
98 
98 
98 
98 
97 
97 
97 
97 
96 
97 
96 
96 
95 
90 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
93 
92 
92 
92 
92 
91 
92 
91 
91 
90 
91 
90 
91 
90 


/ CSC 

10. 


/tan 
9. 


d 

r 


/col 
10. 


/ see 
10. 


d 

I' 
1 

2 
2 

1 
2 
1 
2 
1 
2 
2 
I 
2 
I 
2 
2 
1 
2 
1 
2 
2 
1 
2 
2 
I 
2 
2 
1 
2 
1 
2 
2 
1 
2 
2 
2 
1 
2 
2 
1 
2 
2 
1 
2 
2 
2 
1 
2 
2 
1 
2 
2 
2 
1 
2 
2 
2 
1 
2 
2 
2 

d" 

r 


/ cos 
9. 
99675 

674 
672 
670 
669 


19 
58 
57 
56 
55 
54 
53 
52 
51 

50 

49 
48 
47 
46 


" 


Pr 
105 


oportio 
104 


nalPar 
103 


ts 
102 

"o 

2 
3 
5 


91411 

308 
205 
103 
001 


08914 
09019 
123 
227 
330 


105 
104 
104 
103 


91086 
90981 
877 
773 
670 


00325 
326 
328 
330 
331 




2 
3 



2 
4 
5 



2 
3 
5 



2 
3 
5 


5 

6 

7 
8 
5 


09101 
202 
304 
405 
506 


90899 
798 
696 
595 
494 


434 
537 
640 
742 
845 


103 
103 
102 
103 
102 
102 
101 
102 
101 
101 
101 
101 
100 
100 
100 
100 
9 
99 
9U 
99 
9S 
98 
98 
98 
98 
97 
98 
97 
97 
9(3 
97 
96 
9fl 
96 
9f 
95 
95 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 
93 
91 
92 


566 
463 
360 

258 
155 


333 
334 
336 
337 
339 


667 
666 
664 
663 
661 
659 
658 
656 
655 
653 


5 

6 
7 
8 
9 


9 
10 
12 
14 
Hi 


9 
10 
12 
14 
Hi 


9 
10 
12 
14 
15 


9 
10 
12 
14 
15 


10 

11 
12 
13 
14 


606 
707 
807 
907 
10006 


394 
293 
193 
093 
89994 


947 
10049 
150 
252 
353 


053 
89951 
850 

748 
647 


341 
342 
344 
345 
347 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


18 
19 
21 
23 
24 


17 
19 
21 
23 
24 


17 
19 
21 
22 
24 


17 
19 
20 
22 
24 


15 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 


106 
205 
304 
402 
501 
599 
697 
795 
893 
990 
11087 
184 
281 
377 
474 


894 
795 
696 
598 
499 
401 
303 
205 
107 
Oil) 
88913 
816 
719 
623 
526 


454 
555 
656 
756 
856 
956 
11056 
155 
254 
353 
452 
551 
649 
747 
845 


546 
445 
344 
244 
144 
" " 044 
88944 
845 
746 
647 
548 
449 
351 
253 
155 


349 
350 
352 
353 
355 
~ 357 
358 
360 
302 
363 
365 
367 
368 
370 
371 


651 
650 
648 
647 
645 
643 
042 
640 
638 
637 
635 
633 
632 
630 
029 


45 

44 
43 
42 
41 
40 
3fl 
38 
37 
30 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 


26 
28 
30 
32 
33 
~~35 
37 
38 
40 
42 
44 
46 
47 
49 
51 


26 
28 
29 
31 
33 
35~ 
30 
38 
40 
42 
""43 
45 
47 
49 
50 
52 
54 
55 
57 
59 


26 
27 
29 
31 
33 
34 
Hfl 
38 
39 
41 
"~43 
45 
46 
48 
50 


25 
27 
29 
31 
32 
34 
30 
37 
39 
41 
43 
44 
46 
48 
49 
51 
53 
54 
56 
5R 


30 

31 
32 
33 
34 


11570 
666 
761 
857 
952 


88430 
334 
239 
143 
048 


11943 
12040 
138 
235 
332 


88057 
87960 
862 
765 
668 


00373 

375 

376 
378 
380 


99027 
625 
024 
022 
020 


52 
54 
56 

58 
60 


52 
53 
55 
57 

58 


35 

36 
37 
38 
39 


12047 
142 
236 
331 
425 


87953 
858 
764 
669 
575 


428 
525 
621 
717 
813 


572 
475 
379 
283 
187 


382 
383 
385 

387 
388 


618 
017 
015 
013 
012 


25 

2-1 
23 
22 

2] 

20 

19 
18 
17 
10 


35 

36 
37 
38 
39 


01 
63 
65 
66 
68 


61 
62 
64 
66 
68 


60 
62 
64 
65 
67 


59 
61 
(W 
05 
66 


40 

41 
42 
43 

44 


519 
612 
706 
799 
892 


481 
388 
294 
201 
108 


909 
13004 
099 
194 

289 


091 
86996 
901 
806 
711 


390 
392 
393 
395 
397 


010 
008 
007 
605 
603 


40 

41 
42 
43 
44 


70 
72 
74 
75 
77 


69 
71 
73 
75 
76 


69 
70 
72 

74 
76 


68 
70 
71 
73 
75 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


985 
13078 
171 
263 
355 
447 
539 
630 
722 
813 


015 
86922 
829 
737 
645 
553 
461 
370 
278 
187 


384 

478 
573 
667 
761 


616 
522 
427 
333 
239 


399 
400 
402 
404 
405 


001 
GOO 
598 
596 
595 


15 

14 
13 
12 
11 


45 

40 
47 
48 
49 


79 
80 

82 
84 
86 


78 
80 
81 
83 
85 
87 
88 
90 
92 
94 


77 
79 
81 
82 
84 


77 
78 
80 
82 
83 
85 
87 
88 
90 
92 


854 
948 
14041 
134 
227 


146 
052 
85959 

see 

773 


407 
409 
411 
412 
414 


593 
591 

589 
588 
586 


1( 

1 


50 

51 
52 
53 

54 


88 
89 
91 
93 
94 


86 
88 
89 
91 
93 


55 

56 
57 
58 
59 


904 
994 
14085 
175 
266 


096 
006 
85915 
825 
734 
85644 


320 
412 
504 
597 

688 


680 
588 
496 
403 
312 


416 
418 
419 
421 
423 


584 
582 
581 
579 
577 


i. 


55 

56 
57 
58 
59 


96 
98 
100 
102 
103 


95 
97 
99 
101 
102 


94 
96 
98 
100 
101 


93 
95 
97 
99 
100 


60 


14356 


14780 


85220 


00425 


99575 





60 


105 


104 


103 


102 


/ 


9. 

1 COS 


d 

1' 


10. 

1 80P. 


9. 

/ cot 


d 
1' 


10. 

/tan 


10. 

/ CSC 


9. 

/ sin 


' 


" 


105 

I 


104 

>roporti 


103 

onal Pa 


102 

irts 



97 



82 



50 



TABLE II 





1 

2 
3 


VOL 


2 
3 
5 


100 

"o" 
2 
3 

5 


09 



2 
3 

5 


98 

2 
3 

5 


97 

'() 
2 
3 

5 


Pr< 
96 


>portio 
95 

2 
3 
5 


nal Pai 
94 

2 
3 
5 


rts 
93 


92 


91 


90 

1 
3 
5 
6 


2 


1 






o 



2 
3 

5 



2 
3 
5 

6 




2 

3 
5 

o 



2 
3 
5 
G 
8 
9 
11 
12 
14 







o 


5 

() 
7 
8 
9 
10 
11 
12 
13 
11 
15 
Hi 
17 
18 
19 
20 
2! 
22 
23 
24 
25" 
26 
27 
28 
29 


K 
10 

IS 
15 
17" 
10 
20 
22 
24 
25 
27 
21) 

30 
32 

34 
35 
37 

3<> 

40 
' 42 
14 
45 
47 
4) 
50~ 
52 
51 
5(i 
57 
50 
(il 
02 
<>4 
00 
07 
GO 
71 
72 
74 


"" 8" 
10 
12 
13 
15 
17" 
18 
20 
22 
23 
25 ~ 
27 
28 
30 
32 
33 
35 
37 
38 
40 


~~8~ 
10 

12 
13 

15 
16 
18 
20 
21 
2:1 
25 
26 
28 
30 
31 
33 
35 

3 

3s 
40 


8"" 
10 

11 

13 
15 
16 

18 
20 
21 
23 
24 
20 
28 
29 
31 


8~ 
10 
11 
13 
15 


8~ 
10 

11 

13 
14 


8 
10 

It 

13 
14 

10 

17 

19 
21 

22 


8 
9 
11 
13 

14 


8 
9 
11 
12 

14 


8 

11 
12 

14 


7 

9 
11 

12 
13 
15 
17 
18 
10 
21 
















10 

18 
10 

21 
23 


10 

18 
19 
21 
22 
24 
26 
27 
29 
30 
~32~ 
34 
35 
37 
38 
40 
42 
43 
45 
46 


16 
17 

19 
20 

22 


16 

17 
19 
20 
22 
23 
25 
26 
28 
29 


15 

17 
18 
20 
21 


15 
17 
18 
20 

21 





















24 

20 
27 
29 
31 
32 
34 
36 
37 
39 
40 
42 
44 
45 
47 


24 
25 

27 
28 
30 
32 
33 
35 
36 
38 
40 
41 
43 
44 
40 


23 

25 
27 
28 
30 
31 
33 
34 
30 
38 


23 
25 

20 

28 
29 


23 
24 

20 
27 

20 


23 

24 
25 
27 
29 




1 
1 


33 
34 

30 
38 
39 
41 
42 
44 
46 
47 


31 
33 
34 
36 
37 
39" 
40 
42 
43 
45 


31 

32 
34 
35 

37 
38 
40 
41 
43 
44 


30 

32 
33 
35 
36 


30 
31 
33 
35 
36 
37 
39 
41 
42 
43 









~ 






42 
43 

45 
47 

48 


41 

43 
45 
46 
48 
50 
51 
53 
54 
50 
58 
50 
01 
63 
64 


30 
41 
42 
44 
45 
47 
49 
50 
52 
53 


38 
30 
41 
42 

44 




30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


50 
52 
53 

55 
57 

58 

00 

62 
63 

05 
67 
68 
70 
72 
73 
75~ 
77 
78 
80 
82 
83 
85 
87 
88 
90 


4!) 
51 
52 
54 
56 
57 
59 
60 
02 
64 
65 
07 
69 
70 
72 
73 
75 
77 
78 
80 
82 
83 
85 
87 
88 


48 
50 
52 
53 
55 
57 
58 
00 
61 
03 
65 
66 
08 
70 
71 


48 
50 
51 
53 
54 
50 
58 
59 
61 
62 
4 
66 
67 
GO 
70 


48 
49 
51 
52 

54 


46 

48 
50 
51 
53 
54 
50 
57 
59 
60 


40 
48 
40 
51 
52 


46 

47 
49 
50 
52 


45 
47 

48 
40 

51 







55 

57 
59 
00 
62 


55 
56 

58 
60 
01 


54 

55 
57 

58 
GO 


53 
55 

56 
58 
50 


53 

54 
55 

57 
59 


1 


00 
68 
60 
71 
73 


63 

05 
66 
08 
70 


63 
64 

00 
67 

60 


02 
64 
65 
67 

08 


61 

G3 
64 
66 
67 
GO 
71 
72 
74 
75 
77 
78 
80 
81 
83 


61 

62 
64 
65 
67 


60 
61 
63 
65 

66 


1 
1 
1 
1 
1 





70 
77 
79 
SI 
82 
" 84 
80 
88 
89 
01 


74 

70 
78 
79 
81 
"82 
84 
86 
87 
80 


73 
74 

70 
78 
79 

81 

82 
84 
86 

87 


72 
74 
75 
77 

78 
"80 
82 
83 
85 
86 


71 

73 
74 

70 
78 
" 79 " 
81 
82 
84 
86 


71 

72 
74 

75 
77 

" 78 ~ 
80 
81 
83 
85 


70 
71 

73 
74 

76 
78" 
79 
81 
82 
84 


68 

70 
71 
73 
74 
70 
77 
79 
80 
82 


67 
69 
71 

72 

*L 

75 
77 

78 
79 

81 


2 
2 
2 
2 
2 

2 
2 

2 
2 
2 


55 

56 
57 
58 
59 


03 
94 
90 
08 
09 


02 
03 

05 
07 
08 

100 " 


91 
92 

94 
96 
97 


90 
01 
93 
05 
96 


80 
91 
02 
94 
95 


88 
90 
91 
03 
04 


87 
89 
90 
92 
03 


80 
88 
80 
91 
92 


85 
87 
88 
00 
91 


84 
80 
87 
89 
00 


83 

85 
86 
88 
80 


83 
84 
85 
87 
89 


2 
2 
2 
2 
2 
2 
2 




60 


101 
101 


99 


98 
98 


07 
97 


90 


95 


94 
"94 
onalP 


93 


92 


91 


00 
90 




ft 


100 


9? 


96 

Pi 


95 

roporti 


93 

arts 


92 


91 



TABLE II 



17T 



' 


/ sin 
9. 


d 

1' 

89 
90 
89 
90 
89 
88 
89 
89 
88 
88 
88 
88 
87 
88 
87 
87 
87 
87 
86 
86 
87 
86 
85 
86 
85 
86 
85 
85 
85 
84 
85 
84 
84 
84 
84 
83 
84 
83 
83 
83 
83 
83 
82 
82 
83 
82 
81 
82 
82 
81 
81 
81 
81 
81 
81 
80 
80 
80 
80 
80 


I CSC 

10. 


I tan 
9. 


d 
1' 


I cot 
10. 


I sec 
10. 


d 
1' 


1 COS 

9. 


f 




" 


Proport 
92 


ional Par 
91 


ts 
90 




1 

2 
3 
4 


14356 
445 
535 
624 

714 


85644 
555 
465 
376 
286 


14780 
872 
963 
15054 
145 


92 
91 
91 
91 
91 
91 
90 
91 
90 
90 
89 
90 
89 
90 
89 
89 
88 
89 
88 
88 
88 
88 
88 
87 
88 


85220 
128 
037 
84946 
855 


00425 

426 
428 
430 
432 


1 

2 
2 
2 
2 
1 
2 
2 
2 
2 
1 
2 
2 
2 
2 
2 
1 

2 
2 
2 
2 
2 
1 


99575 
574 
572 
570 

568 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 




1 

2 
3 
4 
5 

6 
7 
8 
9 
10 
11 
12 
13 
14 



2 
3 
5 





2 
3 
5 

6 



1 
3 
5 

G 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 


803 
891 
980 
15069 
157 
245 
333 
421 
508 
596 


197 
109 
020 
84931 
843 


236 
327 
417 
508 
598 


764 
673 
583 
492 
402 


434 
435 

437 
439 

441 


566 
565 
563 
561 
559 


8 
9 
11 
12 

14 
15 
17 
18 

20 

21 


8 
9 
11 
12 
14 


7 

9 
11 

12 
13 

15 
17 

18 

19 

21 
23 
24 
25 

27 

29 

_ __._. 

31 

33 
35 

30 


755 
667 
579 
492 
404 


688 
777 
867 
956 
16046 


312 
223 
133 
044 
83954 


443 
444 
446 
448 
450 


557 
556 
554 
552 
550 


15 
17 

1M 

20 

21 
23 
24 
20 
27 

29 


15 

16 
17 
18 
19 


683 
770 
857 
944 
1G030 


317 
230 
143 
056 
83970 


135 
224 
312 
401 

489 


865 
776 
688 
599 
511 


452 

454 
455 
457 
459 


546 
515 
543 
541 


45 

44 
43 
42 
41 




15 

16 
17 
18 
19 


23 
25 

120 
28 
29 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


116 
203 
289 
374 
460 


884 
797 
711 
626 
540 


577 
665 
753 
841 
928 


423 
335 
247 
159 
072 


461 
463 
465 
467 
468 


539 
537 
535 
533 
532 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 


31 

XI 
34 
35 

:37 


30 

32 
33 
35 
36 


545 
631 
716 
801 
886 
16970 
17055 
139 
223 
307 


455 
369 
284 
199 
114 


17016 
103 
190 
277 
363 


87 
87 
87 
813 
87 
80 
80 
80 
80 
80 
85 
80 
85 
85 
85 
85 
84 
85 
84 
84 
84 
84 
83 
84 
83 
83 
83 
83 
83 
83 
82 
82 
82 
82 
82 


82984 
897 
810 
723 
637 


470 

472 
474 
476 
478 
00480 
482 
483 
485 
487 


2 

2 
2 

2 

1 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
1 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


530 
528 
526 
524 
522 


35 

34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 




25 

26 
27 
28 
29 
30 

at 

32 
33 
34 
35 

36 
37 
38 
39 


38 

40 
41 
43 
44 

~To 

48 
49 
51 

52 


38 
39 
41 
42 
44 
46 
47 
49 
50 
52 


37 

39 
41 
42 
43 
45 
47 
48 
49 
51 


83030 
82945 
861 

777 
693 


17450 
536 
622 
708 
794 


82550 
464 
378 
292 
206 


99520 
518 
517 
515 
513 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


391 
474 
558 
641 
724 


609 
526 
442 
359 

276 


880 
965 
18051 
136 
221 


120 
035 
81949 
864 
779 


489 
491 
493 
495 
497 


511 
509 
507 
505 
503 
501 
499 
497 
495 
494 


54 

55 
57 
58 
00 
61 
03 
64 
(iO 
67 


53 
55 

50 
58 
59 


53 

54 
55 

57 
59 

00 

61 

03 
65 
00 
67 
09 
71 

73 


807 
890 
973 
18055 
137 


193 
110 
027 
81945 
863 


306 
391 
475 
560 
644 


694 
609 
525 
440 
356 


499 
501 
503 
505 
506 


40 

41 
42 
43 
44 


61 

02 
64 
05 
67 
68 
70 
71 
73 
71 


220 
302 
383 
465 
547 


780 
698 
617 
535 
453 
372 
291 
210 
129 
048 


728 
812 
896 
979 
19063 
f46 
229 
312 
395 
478 


272 

188 
104 
021 
80937 


508 
510 
512 
514 
516 


492 
490 
488 
486 
484 


15 

14 
13 
12 
11 


45 

46 
47 
48 
49 
5<f 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


09 
71 

72 
74 

75 
77 

78 
80 

HI 
S3 
84" 
80 
87 
89 
90 


50 

51 
52 
53 
54 


628 
709 
790 
871 
952 
19033 
113 
193 
273 
353 
19433 


854 
771 
688 
605 
522 


518 
520 
522 
524 
526 


482 
480 
478 
476 
474 


10 

9 
8 
7 
6 
5 
4 
3 
2 
1 


70 
77 

79 
80 

82 


75 
77 

78 
79 

81 


55 

56 
57 

58 
59 


80967 
887 
807 
727 
647 


561 
643 
725 
807 
889 


439 
357 
275 
193 
111 


528 
530 
532 
534 
.536 


472 
470 
468 
466 
464 


83 
85 
86 
88 
89 


83 

84 
85 
87 
89 


60 


80567 


19971 


80029 


00538 


99462 





92 


91 


90 


' 


9 

1 COS 


d 

1 


10. 

/ sec 


9. 

I cot 


d 


10. 

tan 


10. 

I CSC 


d 

V 


9. 

/ sin 


/ 


" 


92 

Prop 


91 

ortional I 


90 

>arts 



98 



sr 



62 



TABLE II 



" 


89 


88 


87 


86 


Pi 

85 


roportio 
84 


nal Par 
83 


ts 
82 


81 


80 





1 




1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
10 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 



1 
3 



1 

3 




1 
3 



1 

3 


















1 
3 
















3 


3 


3 


3 


3 



7 

9 
10 

12 
13 
15 
16 
IS 
19 
21 
22 
24 
25 
27 
28 
30 
31 
33 
34 
36 
37 
39 
40 
42 
43 
44 
40 
47 
19 
50 
52 " 
53 
55 
56 
58 
59 
(11 
62 
04 
65 
(17 
OS 
70 
71 
73 


6 



7~~ 
9 
10 
12 
13 
~14 
10 
17 
19 
20 
22 
23 
25 
26 
28 
" ~i;9 
30 
32 
33 
35 
30 
38 
39 
41 
42 





6 


6 


6 


5 


5 


5 


7 

9 
10 

12 

13 
15 

10 

18 

19 
21 

o> 
23 
25 
26 
2S 
29 
31 
32 
34 
35 
37 
38 
40 
41 
43 


7 
9 
10 
11 
13 
"14 
10 
17 
19 
20 


7 

8 
10 

11 

13 
14 
16 
17 
18 
20 
''1 
23 
24 
26 
27 
"28 
30 
31 
33 
34 
35 
37 
38 
40 
41 


7 
8 
10 
11 
_tt 
14 
15 
17 
18 
20 


7 
8 
10 
11 
12 
14 
15 
17 
is 
19 


7 
8 
10 
11 
12 
14 
15 
16 
18 
19 
21 
22 
23 
25 
26 
27 
29 
30 
31 
33 
34 
36 
37 
38 
40 


7 
8 
9 
11 
12 
14 
15 
10 
18 
19 


7 
8 
9 
11 

12 
____ 

15 
10 
17 
19 







1) 




~0 

1 

1 
1 
1 














21 

23 
24 

20 
27 
29 
30 
32 
33 
34 
" "30 
37 
39 
40 
42 


21 
22 
24 
25 
27 
28 
29 
31 
32 
34 
35" 
36 
38 
39 
41 


21 
22 
24 

25 
26 
28 
29 
30 

33 
35 

36 
37 

39 
40 


20 
22 

23 
24 
26 

27 
28 
30 
31 
32 
" 34~ 
35 
36 
38 
39 


20 
21 
23 
24 
25 
~27~ 
28 
29 
31 
32 
~33 
35 
36 
37 
39 













"o 







44 
45 

47 
48 
50 
51 
53 
54 
56 
57 


44 

45 
46 

4S 
49 


43 
44 
40 
47 

49 


42 

44 
45 

47 

4S 


42 
43 

45 
40 
48 
49" 
50 
52 
53 
55 
50 
57 
59 
00 
62 


42 

43 
44 
46 

47 


41 
42 
44 
45 
46 


40 

42 
43 
45 

40 


40 
41 
43 
41 
45 
47 
48 
19 
51 
52 


1 
1 
1 
I 




1 
1 
1 
1 


51 
52 
54 
55 
57 


50 
52 
53 
54 

50 


50 

51 
52 
54 
55 
~57 
58 
60 
01 
62 


48 
50 
51 
53 

54 


48 
49 
51 

52 
53 


47 
49 
50 
51 
53 


1 
1 
1 
1 
1 
" "~i 
i 
i 
i 
i 


1 
1 
1 

1 
~1 

1 
1 
1 
1 


59 

(50 
62 
03 
65 
00 
67 
09 
70 
72 
73 
75 
76 
7S 
79 


58 
59 
01 
62 

04 


57 

59 
00 
62 

03 


55 

57 
58 
59 
61 


55 

56 
57 

59 
60 


5t 
55 

57 
58 
59 


53 

55 
50 
57 
59 


05 
67 
08 
70 
71 
~~W~ 
74 
75 
77 
78 


65 

00 
67 

09 

70 


04 
05 
67 
08 
69 


(53 
64 
(50 
07 
69 
~~70~ 
71 
73 
74 
76 


02 
64 
65 
66 

08 


61 

63 
64 
66 

67 


61 
02 
63 

65 
66 


60 

61 

03 
64 
65 
67 
68 
69 
71 
72 


2 
2 
2 
2 

o 
2 
2 
2 
2 


1 
1 
1 

1 
1 __ 
1 
1 

1 

1 
1 

1 
1 
1 
1 


74 
76 

77 
79 

80 
82 
83 
85 
86 
88 


72 

73 
75 
70 
77 


71 
72 
74 
75 
76 
78 
79 
81 
82 
84 


09 
71 
72 
73 

75 


68 

70 
71 

72 
74 


08 

69 
70 
72 
73 
74~ 
76 
77 
78 
80 


55 

56 
57 
58 
59 


81 
S2 
84 
85 
87 


so 
si 
83 
84 
86 


79 
80 
82 
S3 
85 
86 
~ 86" 


77 
78 
80 
81 
83 


76 
77 

79 
80 
82 


75 
77 
78 
79 
81 


73 
75 

76 
77 
79 


2 
2 
2 
2 
2 


60 


89 
89 


88 


87 


85 


84 


83 


82 


81 


80 


2 


1 


ff 


88 


87 


85 
Pr 


84 
oportioc 


83 

ial Part 


82 

5 


81 


80 


2 


1 



53 



TABLE II 



170 C 



/ 


I sin 
9. 


d 

l' 


I CSC 

10. 


Z tan 
9. 


d 
1' 


I cot 
10. 


I sec 
10. 


d 

l' 


I COS 

9. 


r 




// 


82 


81 


801 


1- 
79 


*0] 

78 


aor 
77 


tlOl 

76 


lal 
75 


Pa 
74 


rts 
73 


721 


71 


3 


2 




1 

2 
3 
4 


19433 
513 
592 
672 
751 


80 
79 
80 
79 

79 


80567 

487 
408 
328 
249 


19971 
20053 
134 
216 

297 


82 
81 
82 
81 
81 
81 
81 
81 
80 
81 
80 
80 
80 
80 
80 
79 
80 
79 
79 
79 
79 
79 
78 
79 
78 
78 
78 
78 
78 
78 
77 
78 
77 
77 
77 
77 
77 
76 
77 
76 
76 
77 
76 
76 
75 
76 
75 
76 
75 
75 
75 
75 
75 
74 
75 
74 



74 
74 

d 
1' 


80029 
79947 
866 
784 
703 


00538 
540 
542 
544 
546 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 

2 
2 
3 
2 

2 
2 
1 

2 
2 
3 

2 
2 
2 
2 
2 

2 
3 
2 

2 


99462 
460 
458 
456 
454 


60 

59 
58 
57 
56 




1 

2 
3 

4 



1 
3 
4 
5 



1 

3 
4 
5 



1 
3 



1 
3 



1 
3 



1 
3 



1 
3 



1 
2 



1 
2 



1 
2 




1 

2 



1 
2 
















5 


5 


5 


5 


5 


5 


5 


5 


5 


5 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


830 
909 
988 
20067 
145 


79 
79 

79 
78 
78 
79 
78 
78 
77 
78 
78 
77 
77 
77 
77 
77 
77 
76 
77 
76 
76 
76 
76 
75 
76 
75 
76 
75 
75 
75 
74 
75 
75 
74 
74 
74 
74 
74 
74 
73 
74 
73 
73 
73 
73 
73 
73 
72 
73 
72 
72 
73 
71 
72 
72 


170 
091 
012 
79933 
855 
777 
698 
620 
542 
465 
387 
309 
232 
155 
078 


378 
459 
540 
621 
701 
782 
862 
942 
21022 
102 


622 
541 
460 
379 
299 
~~2l8 
138 
058 
78978 
898 


548 
550 
552 
554 
556 
~~558 
560 
562 
564 
566 


452 
450 
448 
440 
444 
442 
440 
438 
430 
434 


55 

54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 


5 

6 

7 
8 
c 

To 

11 

12 
13 
14 


7 
8 
10 

11 
12 
14 

15 
16 

18 
19 


7 
8 
9 
11 
12 
14 
15 
Ifi 
18 
19 
20 
22 
23 
24 
26 


7 
8 
g 

11 

12 


7 
8 
9 
11 

12 


7 

8 
9 
10 
12 
13 
14 
16 
17 
18 
19 
21 
22 
23 
25 

20 
>7 

29 

30 
31 


6 

8 
9 
10 
12 



8 
9 
10 
11 
13 
14 
15 
16 
18 
19 
20 
22 

24 


6 
8 
9 

10 
11 
12 

14 
15 
10 
18 
19 
20 
21 
22 
24 


6 
7 

9 
10 
11 



7 
9 
10 
11 


6 
7 
8 
10 
11 


6 
7 
8 
9 
11 









1 
1 

1 
1 
1 
1 
1 
1 
1 














223 
302 
380 
458 
535 


13 

15 
10 
17 
19 


13 
14 

10 
17 
18 


13 
14 
15 
17 
18 
19 
21 
22 
23 
24 


12 
14 

15 
10 
17 
19 
20 
21 
22 
23 


12 
13 
15 

16 
17 

18 

19 

22 
23 


12 
13 
14 

10 
17 


12 
13 
14 
15 
17 
18 
19 
20 
21 
22 


613 
691 
768 
845 
922 


182 
261 
341 
420 
499 


818 
739 
659 
580 
501 


568 
571 
573 
575 
577 


432 
429 
427 
425 
423 


15 

10 
17 
18 
19 


21 

*>ij 
23 
25 

20 


20 
21 
23 
24 
25 
27 
2S 
29 
31 
32 
33 
35 
3( 
37 
31 


20 
21 
22 

24 
25 
20 

28 
29 
30 
32 


18 
19 

:, 

23 
24 
25 
26 

28 
2(1 



1 


20 

21 
22 
23 
24 


999 
21076 
153 
229 
306 


001 
78924 
847 
771 
694 


578 
657 
736 
814 
893 


422 
343 
264 
186 
107 


579 
581 
583 

585 
587 


421 
419 
417 
415 
413 


40 

39 
38 
37 
30 


20 

21 
22 
23 
24 
25 
2( 
27 
28 
29 


27 

29 
30 
31 
33 


27 
28 
30 
31 
32 


26 
27 
28 
30 
31 


25 
27 

28 
29 
30 
32 
33 
34 
35 
37 
38 
39 
41 
42 
43 
44 
46 
47 
48 
49 


25 
20 
28 
29 
30 
31 
32 
34 
35 
30 
38 
39 

t? 

2 

45 
4< 
48 
49 


25 
20 
27 
28 
30 
:ri 
32 
33 
35 
30 
37 
38 
39 
41 
42 
43 
44 
40 
47 
48 


24 
26 
27 
28 
29 
30 
32 
33 
34 
35 
36 
38 
39 
40 
41 
43 
44 
45 
10 
47 


24 
25 

20 
27 

28 


1 

1 
1 
1 
1 
~1 
1 
1 
1 
1 

~2 
2 
2 

2 
~2 
2 
2 

2 
"2 

2 
2 
2 
2 


25 

26 
27 
28 
29 


382 
458 
534 
610 
685 


618 
542 
466 
390 
315 


971 
22049 
127 
205 
283 


029 
77951 
873 
795 
717 
77639 
562 
484 
407 
330 
253 
17( 
099 
023 
7694( 


5S9 
591 
593 
596 
598 


411 
409 
407 
404 
402 


35 

34 
33 
32 
31 




34 
36 

:i7 

3R 
40 


34 
35 
36 
38 
39 
40 
42 
43 
45 
4< 


33 
34 
36 
37 
38 
40 
41 
42 
43 
45 


33 

34 
35 
36 
38 
39 
40 
42 
43 
44 


32 
33 

35 
30 
37 
38 
4( 
41 
42 
44 


30 
31 
32 

34 

35 
30 

38 

40 

42 
43 
4-4 
46 

47 


30 

31 
32 
33 

:w 
36 

37 
38 
39 
40 
41 
43 
44 
45 
4( 




30 

31 
32 
33 
34 


21761 
836 
912 
987 
22062 


78239 
164 
088 
013 
77938 


22361 
438 
516 
593 
670 


00600 
002 
604 
606 
608 
610 
612 
615 
617 
019 


99400 
398 
390 
394 
392 
390 
388 
385 
383 
381 


30 

29 
28 
27 
2f 
25 
24 
23 
22 
21 




30 

31 
32 
33 
34 


41 
42 
44 
45 
46 


4( 
41 
43 
44 
45 
47 
4S 
4! 
51 
52 


J 

"j 


35 

36 
37 
38 
39 


137 
211 
286 
361 
435 


863 
789 
714 
639 
565 
"491 
417 
343 
269 
195 


747 
824 
901 
977 
23054 


35 

3f 
37 

38 
3< 


48 
49 
51 
52 
53 


11 

50 
51 
53 


4f 
47 
49 
50 
51 


45 

17 
48 
49 
51 


45 
41 
47 

4< 

5( 


40 

41 
42 
43 
44 


509 
583 
657 
731 
805 


130 
206 
283 
359 
435 


870 
794 
717 
641 
565 


621 
023 
625 
028 
030 


2 
2 

a 

2 
2 
2 
2 
2 
3 
2 
2 
2 
2 
3 
2 
2 
2 
2 
3 
2 


379 
377 
375 
372 
370 


20 

19 
18 
17 
]( 




40 

41 
12 
43 
44 


55 

5' 
57 

59 
(JO 
61 
63 
(34 
66 
07 
68 
70 
71 
72 
74 
75 
77 
78 
79 
81 


54 
55 

57 
58 
59 


53 

55 
50 
57 
59 
60 
61 
03 
04 
05 
67 
08 
69 
71 
72 


53 
54 
55 
51 

58 


52 
53 
55 

50 
57 


51 
53 

54 
55 
56 


51 
52 
53 
54 
56 


5f 
51 
52 

54 
55 


49 
51 

52 
53 
54 


49 48 
50', 49 
5150 
52152 
545: 


47 
49 

5( 
51 
52 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


878 
952 
23025 
098 
171 
" 244 
317 
390 
462 
535 


122 
048 
76975 
902 
829 
"766 
683 
610 
538 
465 


510 
586 
661 
737 
812 


490 
414 
339 
263 
188 
"113 
038 
75963 
888 
814 


032 
034 
036 
038 
041 
043 
045 
047 
049 
052 


308 
300 
304 
302 
359 
357 
355 
353 
351 
348 


L; 
13 
12 
11 

io 

9 

8 

6 


45 

4( 
47 
48 
49 
50 
51 
52 
53 
54 


61 
02 
63 

05 
00 


59 
61 

02 
03 
65 
06 
07 
68 
70 
71 
72 
74 
75 
76 
78 


59 

60 
61 
62 

04 
05 
00 
68 

69 
70 


58 
59 
60 
62 
63 
64 
65 
67 
68 
69 
71 
72 
73 
74 
76 


57 
58 
60 
61 
02 
63 
65 
60 
67 
68 


56 
58 
59 
60 
61 
62 
64 
05 
66 
68 


55 

57 
58 
59 
60 


55 
50 
57 
58 
00 


54 
55 
56 
58 
59 
"66 
61 
62 
64 
65 


53 
54 

5f 
57 
58 
5! 
60 
62 
63 
04 


2 
2 
2 
2 
2 
2 
3 
3 
3 
3 


-^ 


887 
962 
24037 
112 
186 


68 

09 
70 
72 

73 


02 
63 
04 
05 
67 
68 
69 
70 
72 
73 


01 
02 
63 
64 

66 


55 

56 
57 
58 
59 


607 
679 
752 
823 
895 


593 
321 
248 
177 
105 


261 
335 
410 
484 
558 


739 
065 
590 
516 
442 


054 
056 
058 
600 
663 


340 
344 
342 
340 
337 


5 



] 


55 

56 
57 

58 
59 


74 
76 

77 
78 
80 


73 
75 

70 
77 

79 


71 

73 
74 
75 

77 


70 
71 
72 
73 
75 


69 
70 
71 
72 

74 


67 
68 
69 
71 

72 


66 
67 
68 
70 

71 


65 
66 
67 

69 
70 


3 
3 
3 
3 
3 




60 


23967 


76033 


24632 


75368 


00605 




99335 





60 


82 


81 


80 


79 


78 


77 


76 


75 


74 


73 


72 


71 


3 


2 

X 


' 


9. 

f cos 


d 
1' 


10. 

I sec 


9. 

Jcot 


10. 

Ztan 


10. 

I CSC 


d 


9. 

1 sin 


/ 






82 


81 


80 


79 

I 


78 
*0 


77 

poi 


76 

tio 


75 

nal 


74 

Pi 


73 

irti 


72 

s 


71 


3 



99 C 



80 

54 



10 C 



TABLE II 



169 C 



, UWl 

9. 


I C8C 
10. 

76033 
5 75961 
890 
819 
J 747 


tan 
9. 


/cot 
10. 


SPC 

10. 


/ COS / 

9. 





4 


73 


72 


^ro 


% 


tion 


al J 


t>ar 
67 


ts 
66 

1 
2 
3 
4 
5 


65 


3 2 


23967 
1 24039 
2 110 
3 181 
4 253 


4632 
706 
779 
853 
926 


75368 
1 294 
* 221 
1 147 
' 074 
1 000 
74927 
' 854 
* 781 
* 708 


10665 
667 
(569 
672 
674 
676 
678 
681 
683 
685 


99335 60 
333 5C 

331 58 
328 57 
32656 




1 
2 
3 
4 



1 
2 

4 
5 



1 
2 
4 
5 
6 



1 
2 
4 
5 
6 




1 

2 
4 
5 



1 
2 
3 

5 
~6 



1 
2 
3 
5 
6 



1 
2 
3 
5 
6 




l 

2 
3 
4 

G 




1 

2 
3 

4 
5 














5 324 

6 395 
7 466 
8 536 
9 607 


676 
1 605 
534 
3 464 
1 393 


5000 
073 
146 
219 
292 


32455 
32254 
31953 
31752 
31551 


5 

6 

7 
8 
9 


6 


6 


9 
10 
11 


9 

10 
11 


8 
10 

11 


S 
9 
11 


8 
9 
11 


8 
9 
10 


8 
9 
10 


8 
9 
10 


8 
9 
10 


8 
9 
10 


10 677 

11 748 
12 818 
13 888 
14 958 


B 323 
1 252 
182 
& 112 
042 
74972 
902 
832 
> 763 
693 
9 "624 
3 555 
9 486 
9 417 
9 348 


365 
437 
510 
582 
655 
727 
799 
871 
943 
6015 


5 635 
2 563 
3 490 
2 418 
* 345 
2 " 273 
2 201 
2 129 
2 057 
2 73985 


687 
690 
692 
694 
696 


31350 

310 48 
30848 
306 47 
304 46 
'30145 
29944 
297 43 
29442 
29241 




2 
13 
14 


12 
14 

r 

l(! 

[7 

20 
21 

o 

23 

20 
27 
28 
30 


12 
13 
15 

16 
17 
18 
19 

21 
22 
23 


12 
13 
14 

IB 
17 


12 
13 
14 
15 
17 


12 
13 
14 
15 
16 


12 

13 
14 
15 

16 


11 
12 

14 
15 
16 
17 
18 
19 
20 
22 


11 
12 
13 
15 

16 


11 
12 
13 
14 
15 
17 
18 
19 
20 
_21 
22 
23 
24 
25 
26 


11 
12 
13 
14 
15 



1 
1 
1 
1 


15 25028 
16 098 
17 168 
18 237 
19 307 


699 
701 
703 
706 
708 


15 

16 
[7 
18 
19 
20 
21 
22 
2c 
2-4 


18 
19 
20 
22 

23 


IS 
19 
20 
21 
22 
24 
25 
2G 
27 
28 


17 

19 
20 

21 
22 


17 
18 
20 

21 
22 


17 
18 
19 
20 
21 
22 
23 
25 
26 
27 


16 
17 
18 


21 
22 
23 
24 
25 
26 





vO 376 

21 445 
22 514 
23 583 
24 652 


086 
158 
229 
301 
372 


1 914 
2 842 
i 771 

2 699 
i 628 


710 
712 
715 
717 
719 
722 
724 
726 
729 
731 


' 290 40 
J 288 39 
J 285 38 
J 283 37 
J 281 36 


24 
26 

27 
28 
29 


24 
25 
26 

28 
29 


23 
25 

26 

27 
28 


23 
24 
25 
26 

28 


23 

25 

26 
27 


; 


25 721 

26 790 
27 858 
28 927 
29 995 


J 279 
9 210 
8 142 
9 073 
8 005 


443 
514 
585 
655 
726 
26797 
867 
937 
27008 
078 


1 557 
1 486 
1 415 
345 
1 274 
1 73203 
133 
o 063 
i 72992 
922 


' 27835 
* 27634 
* 274 33 
* 271 32 
2 26931 


25 

26 
27 
28 
2< 

30 

31 
32 
33 
34 


31 
32 
*3 
35 
36 
37 
38 
39 
41 
42 


30 

32 
33 
34 
35 
36 
38 
3! 
40 
41 


30 
31 
32 
34 
35 


30 

31 
32 
33 
34 
36 
37 
3S 
39 
40 


29 
30 
31 
3: 
34 
35 
3C 
37 
39 
4( 


29 
30 
31 
32 
33 


28 
29 
31 
32 
33 
34 
35 

3e 

37 
39 


28 
29 
30 
31 
32 
34 
35 
36 
37 
38 


29 
30 
31 
32 


27 
28 
29 
30 
31 
32" 
34 
35 
36 
37 


1 1 
1 1 
1 1 
1 1 
1 1 


,10 26063 
31 131 
32 199 
33 267 
34 335 


* 7J937 
8 869 
8 801 
18 733 
8 665 


00733 
73K 

738 
740 
743 


2 99267 30 

' 264 29 
2 262 28 
2 26027 
3 257 26 


36 
37 

38 
40 
41 


34 

3G 
37 
38 
39 


33 
34 
35 
3G 
37 


2 1 

2 1 
2 1 
2 1 
2 1 


35 403 

36 470 
37 538 
38 605 
39 672 


8 597 
7 530 
18 462 
7 395 
7 328 
7 261 
7 194 
7 127 
> 7 060 
> 7 72993 


148 
218 
288 
357 
427 


852 
o 782 
712 
9 643 
573 
9 "-504 
434 
9 365 
> 9 296 
> 9 227 


745 
748 
750 
752 
755 


- 255 25 

* 252 24 
2 250 23 
2 248 22 
3 245 21 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
4C 
47 
48 
49 


4; 
44 

4f 
47 
48 


43 

44 
45 
4f> 
47 
49 
50 
5 
52 
54 


42 
43 
44 
46 
47 
48 
4 
50 
5 
5 
54 
5 
5 
5 
5 
G 
6 
6 
64 
GJ 
6 
G 
& 
7< 
7 


41 

43 
44 
45 

46 
"47 

49 

50 
51 
52 
53 
54 
5G 
57 
58 
59 
GO 
62 
G3 
64 
~G 
6C 
67 
G 
7C 


4 
4L 
43 
44 
45 
47 
48 
49 
60 
5 
53 
54 
5 
5 
5 
5, 
5 
G 
G 
6 
~G4 
6 
6 
68 
6 


4 
4 
43 

44 
4 
~4 
4 
4 
4 
5 
5 
5 
5 
5 
5 
5 
5 
6 
6 
6 
6 
64 
6 
6 
6 


40 
41 
42 
43 
44 
45 
46 
48 
49 
50 
5 
52 
5. 
54 
56 


39 
40 
41 
42 
44 
45 
40 
47 
48 
49 
50 
51 
52 
54 
55 
56 
57 
58 
59 
60 


39 

40 
4 
4- 
4. 


38 
39 
40 
41 
42 
43 
44 
46 
47 
48 
49 
50 
51 
52 
53 


2 1 
2 1 
2 1 
2 1 
2 1 

2 
2 1 
2 1 
2 1 
2 2 
2 2 
2 2 
2 't 
2 2 
2 2 
3 2 
3 5 
3 2 
3 2 


40 739 

41 806 
42 873 
43 940 
44 27007 


496 
566 
635 
704 
773 


757 
759 
762 
764 
767 


2 24320 
2 241 19 
3 238 18 
2 236 17 
3 233 16 
2 - 231 5 
2 229 4 
3 226 3 
2 224 
* 221 


49 
51 

52 
53 
54 
55 
5 
58 
5 
60 
6 
6 
6 
6 
6 


44 
4 
4 
4 
4 
4 
5 
5 
5 
54 
5 
5 
5 
5 
5 


45 073 

46 140 
47 206 
48 273 
49 339 


10 927 
> 7 860 
* 794 
> 7 727 
5e 66 


842 
911 
980 
28049 
117 


)0 158 
> 9 089 
020 
59 7195 
883 


769 
771 
774 

77(5 
779 


55 
5 
5 
58 
G 
G 
6 
6 
64 
6 
G 
68 
G 
7 
7 


50 405 
51 471 
52 537 
53 602 
54 668 


Jb 595 
* 52 

Mi 4(j 

55 39 
56 33 


186 
254 
323 
391 
459 


jy 81 
J8 74 

67 
60 
58 54 


781 
783 
786 
788 
791 


2 219 1 
2 217 
3 214 
2 212 
3 209 


50 

5 
52 
53 
54 


5 
58 
5 
6 
G 


54 
55 
56 
57 
58 


55 734 

56 799 
57 864 
58 930 
59 995 


96 26 

55 20 

35 13 
w 07 

&5 00 

36 7194 


527 
595 
662 
730 

798 


u* 47 

38 40 

w 33 

58 2 7 

ss 20 


793 
796 
798 
800 
803 


2 207 " 
3 204 
2 202 
2 200 
3 197 


55 

5 
5 
5 
5 


68 
6 
7 
7 

7 


G 
6 

6 
6 
6 


6 
69 

64 
65 
6 


6 

6 
6 
64 
6 


60 
61 
62 
63 
64 


3 2 

3 \ 
3 2 
3 2 


60 28060 


28865 


W 7113 


00805 


2 99195 


6 


7 


7 


7 


71 


7 


G 


G 


6 


6 


65 
MJ5 


3 2 


, 9. 

/cos 


d 10. 

r / sec 


9. 

I cot 


d 10. 

r /tai 


10. 

/CSC 


d 9. 

r I sin 


" 


7 


7 


7 


71 
Pi 


7 

cop 


6 

orti 


6 

ona 


IP 


6 

arts 


3 2 


100 79 

55 



ir 



TABLE II 



168 C 



' 


1 Sill 

9. 


d 

1' 


/ CSC 

10. 


/tan 
9. 


d 

i' 


/cot 
10. 

71135 
067 
000 
70933 
866 
799 
732 
665 
598 
532 


/sec 
10. 


d 

1' 


/cos 
9. 


' 




,/ 


68 


67 


66 


Pr 
65 


opo 
64 


rtic 
63 


nal 
6 


Pa 
61 


rts 
60 


1 
2 
3 


59 


3 













J 















1 

2 
3 

4 
5 

6 

7 
8 
9 


8060 
125 
190 
254 
319 
"384 
448 
512 
577 
641 


05 
05 
04 
65 
05 
04 
04 
05 
64 
64 
04 
04 
63 
64 
64 
63 
63 
64 
03 
63 
63 
63 
63 
62 
63 
62 
63 
62 
62 
63 
62 
62 
61 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
61 
60 
60 
60 
60 
59 
60 
60 
59 
60 


71940 
875 
8JO 
746 
681 
"616 
552 
488 
423 
359 


8865 
933 
9000 
067 
134 
201 
268 
335 
402 
468 


68 
67 
67 
57 
07 
67 
67 
67 
00 
67 
66 
67 
66 
it) 
60 
GO 
GO 
66 
66 
65 
66 
>5 
35 
66 
35 
65 
65 
65 
65 
64 
65 
64 
65 
64 
64 
65 
64 
64 
64 
64 

63 

64 

63 

64 
63 
64 
63 
63 
63 
63 
63 
E13 
63 
62 
63 
62 
63 
62 
62 
62 


00805 
808 
810 
813 
815 
"818 
820 
823 
825 
828 


3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
2 
3 
3 
2 
3 
2 
3 
2 
3 
3 
2 
3 
2 
3 
3 

3 
2 
3 
3 
2 
3 
3 
2 
3 
3 
2 
3 
3 
2 
3 
3 
2 
3 
3 


99195 
192 
190 
187 
185 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 




1 

2 
3 



1 
2 
3 



1 
2 
3 




1 

3 




1 

2 
3 




1 

2 
3 




] 

2 
3 



1 
2 
3 



1 
2 
3 




1 
2 
3 


182 
180 
177 
175 
172 


5 

6 

7 
8 
9 


~0 
7 
8 
9 
10 


~6 
7 
8 
9 
10 


~& 

7 
8 
9 
10 


5 
6 

8 
9 
10 


5 
6 
7 
9 

10 


~5 
6 

7 
8 
9 


5 

6 
7 
8 
9 


~5 
6 

7 
8 
9 


~7> 

6 

7 
8 
9 


5 

6 

7 
8 
9 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

21 
22 
23 
24 


705 
769 
833 
896 
960 
9024 
087 
150 
214 
277 


295 
231 
167 
104 
040 
70976 
913 
850 
786 
723 


535 
601 
668 
734 
800 
866 
932 
998 
30064 
130 


465 
399 
332 
266 
200 
134 
068 
002 
69936 
870 


830 
833 
835 
838 
840 
"843 
845 
848 
850 
853 


170 
167 
165 
162 
160 
157 
155 
152 
150 
147 


50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

21 
22 
23 
24 
5 
26 
27 
28 
20 


11 
1 

14 
15 
10 
17 
IS 
19 
20 



11 
12 
13 
15 

16 
If 
18 
19 
20 
21 
22 
3 
25 
26 
27 


11 
12 
13 
14 
15 
17 
18 
19 
20 
21 
22 
23 
24 
25 
20 


11 
12 
13 
14 
15 
16 
17 
18 

21 
22 
23 
24 
25 
26 


11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
3 
25 
26 
~27 
28 
29 
30 
31 


10 

12 
13 
14 
15 

l(i 

17 

18 
19 
20 
21 
22 
23 
24 
25 
'26 
27 
28 
29 
30 


10 
11 
12 
13 
14 
15 
17 
18 
19 
20 
21 
22 
23 
21 
25 


10 
11 
12 
13 
14 

Ts 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
32 
33 
34 
35 
"36 
37 
38 
39 
40 


10 
11 
12 
13 
14 

Is 

16 
17 
18 
19 
20 
21 
22 
23 
24 


10 
11 
12 
13 
14 
15" 
16 
17 
18 
19 




1 
1 
1 
1 
1 
1 
1 
1 
1 








~0 

1 
1 
1 

1 
__ 

1 
1 
1 
1 
1 
1 
1 
1 
1 
1 

] 
1 


340 
403 
466 
529 
591 


660 
597 
534 
471 
409 


195 
261 
326 
391 
457 


805 
739 
674 
609 
543 


855 
858 
860 
803 
865 


145 
142 
140 
137 
135 


23 
24 
25 
20 
27 


20 
21 
22 
23 

24 


1 
1 
1 
1 
1 

"r 

1 

1 
1 
1 


5 

26 
27 
28 
29 


654 
716 
779 
841 
903 


346 
284 
221 
159 
097 


522 

587 
652 
717 
782 


478 
413 
348 
283 
218 


868 
870 
873 
876 
878 
00881 
883 
886 
888 
891 


132 
130 
127 
124 
122 


28 
9 

31 
32 
33 


28 
29 
30 
31 
32 
34 
35 
36 
37 
38 


7 

29 
30 
31 
32 
"33 
34 
35 
36 
37 
39 
40 
41 
42 
43 


27 
28 
29 
30 
31 
3 
34 
35 
36 
37 


26 
27 
28 
29 
30 


25 

20 
27 

28 
29 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


9966 
30028 
090 
151 
213 
275 
336 
398 
459 
521 


70034 
69972 

910 

849 
787 


30846 
911 
975 
31040 
104 


69154 

089 
025 
68960 
896 
832 
767 
703 
639 
575 


99119 
117 
114 
112 
109 


30 

2f 

28 
27 
26 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


34 
35 
30 
37 
39 


32 
33 
34 
35 
36 


3 

33 
34 
35 
30 


31 
32 
33 
34 
35 


30 
31 
32 
33 
31 


30 
30 

31 
32 
33 


2 
2 
2 
2 
2 


35 

36 
37 

38 
39 


725 
664 
602 
541 
479 


168 
233 
297 
361 
425 
489 
552 
616 
679 
743 


894 
896 
899 
901 
904 


106 
104 
101 
099 
096 


5 

24 
23 
22 
21 


40 
41 
42 
43 
44 
45 
46 
48 
49 
50 


39 
40 
41 
42 
44 


38 
39 
40 
41 
42 


37 
38 
39 
41 
42 


37 
3S 
39 
40 
41 


36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
47 
48 
49 
50 
51 


35 
36 
37 
38 
39 


34 
35 
36 
37 
38 


2 
2 

2 
2 


1 
1 
1 

1 
1 


10 

41 
42 
43 
44 


582 
643 
704 
765 
826 


418 
357 
296 
235 
174 


511 
448 
384 
321 
257 


907 
909 
912 
914 
917 


093 
091 
088 
086 
083 




IS 
18 
17 
1C 
15 
14 
13 
12 
11 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 
~50 
51 
5 
54 
55 


44 
45 
46 
47 
4H 
49 
51 
52 
53 
54 


43 
44 
46 

47 

48 


43 

44 
45 
46 
47 
48 
49 
50 
51 
52 


42 
4,'i 
41 
45 
40 


41 
-12 
43 
44 
45 


40 
41 
42 
43 
44 
~45 
4( 
47 
48 
4f 
~60 
51 
5L 
5: 
54 


39 
40 
41 
42 
43 




2 

2 
2 
2 

T 

2 
2 
2 

2 


1 
1 
1 


45 

46 
47 
48 
49 


887 
947 
31008 
068 
129 


113 
053 
68992 
932 
871 


806 
870 
933 
996 
32059 


194 
130 
067 
004 
67941 


920 
922 
925 
928 
930 


080 
078 
075 
072 
070 


45 

46 
47 
48 
49 


51 
52 
53 
54 
56 


49 
50 
51 
52 
53 


47 

48 
41 
50 
51 


46 
47 
48 
49 
50 


44 
45 
46 
47 
48 
49~ 
50 
51 
52 
53 


50 

51 
52 
53 
54 


189 
250 
310 
370 
430 


811 
750 
690 
630 
570 


122 
185 
248 
311 
373 


878 
815 
752 
689 
627 


933 
936 
938 
941 
944 


067 
064 
062 
059 
056 


10 

g 


7 
6 


50 

51 
52 
53 
64 
55 
56 
57 
58 
59 
60 


57 

58 
59 
GO 
61 


50 
57 
58 
59 
60 


55 
56 
57 
58 
59 
61 
62 
63 
64 
65 
"66 


54 
55 
56 
57 
58 
60 
61 
62 
63 
64 
"65 


53 
54 
55 

57 
58 
"59 
60 
61 
62 
63 
"64 


5 

54 
55 
56 
57 


52 
53 
54 
55 

56 


51 
52 
53 
54 
55 


2 
3 
3 
3 
3 




55 

56 
57 
58 
59 


490 
549 
609 
669 

728 
31788 


510 
451 
391 
331 
272 


436 
498 
561 
623 
685 


564 
502 
439 
377 
315 


946 
949 
952 
954 
957 


054 
051 
048 
046 
043 
99040 


5 

4. 
3 

r 
& 




62 
63 

G5 
60 
G7 


61 
63 

64 
65 
66 


58 
5! 
60 
01 
62 
63 


57 
58 
59 
GO 
61 
62 


56 
57 
58 
59 
60 
~61 


55 
5f 
57 
58 
5* 
60 


54 
55 
56 
57 
58 


3 
3 
3 
3 
3 
3 


-jj 


60 


68212 


3747 


67253 


00960 


68 


67 


59 


' 


9. 

1 COS 


d 
1' 


10. 

2 sec 


9. 

I cot 


d 


10. 

Han 


10. 

Zcsc 


d 

r 


9. 

I sin 


/ 




// 


68 


67 


66 


65 

Pr 


64 

opo 


63 

rtic 


62 

>nal 


61 

Pa 


60 

rts 


59 


3 


101 78 

56 



12 C 



TABLE II 



167 C 



' 


I sin 
9. 


d 

1' 

69 
60 
59 
59 
59 
59 
59 
59 
58 
59 
59 
58 
58 
59 
58 
58 
58 
58 
58 
58 
58 
57 
58 
57 
58 
57 
57 
58 
57 
57 
57 
56 
57 
57 
57 
56 
57 
56 
56 
57 
56 
.56 
56 
56 
56 
56 
55 
56 
55 
56 
55 
56 
55 
55 
55 
55 
55 
55 
55 
55 

"d 
1' 


I CSC 

10. 


I tan 
9. 


d 
1' 

63 
62 
61 
62 
62 
62 
61 
62 
61 
62 
61 
61 
61 
61 
61 
61 
61 
61 
60 
61 
60 
61 
60 
60 
61 
60 
60 
60 
60 
60 
59 
60 
60 
59 
60 
59 
59 
59 
60 
59 
59 
59 
59 
58 
59 
59 
58 
59 
58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
57 

d" 

1' 


I cot 
10. 


/ sec 
10. 


d 


1 COS 

9. 


f 




// 


63 


62 


J 
61 


Pro 
60 


port 
59 


ioni 

58 


il P 
57 


arts 
56 


551 


3 


2 




1 

2 
3 

4 


31788 
847 
907 
966 
32025 


68212 
153 
093 
034 
67975 


32747 
810 
872 
933 
995 


67253 
190 

128 
067 
005 


00960 
962 
965 
968 
970 


2 
3 
3 
2 
3 
3 
2 
3 
3 
3 

2 
3 

3 
2 
3 
3 
3 
2 
3 
3 
3 
2 
3 
3 
3 
2 
3 
3 
3 
3 
2 
3 
3 
3 
3 
3 
2 
3 
3 
3 
3 
3 
2 
3 
3 
3 
3 
3 
3 
3 
2 
3 
3 
3 
3 
3 
3 
3 
3 


99040 
038 
035 
032 
030 


60 

59 
58 
57 
56 






1 

2 
3 

4 



1 
2 
3 

4 



1 
2 
3 

4 



1 
2 
3 
4 



1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 



1 
2 
3 
4 
~5 
6 
7 
8 
9 
10 
11 
12 
13 
14 

"is 

16 
17 
18 
19 




1 
2 
3 
4 



1 
2 
3 

4 



1 
2 
3 
4 



1 
2 
3 

4 
















5 

6 
7 
8 
9 


084 
143 
202 
261 
319 


916 
857 
798 
739 
681 
622 
563 
505 
447 
388 
330 
272 
214 
156 
098 


33057 
119 
180 
242 
303 
"365 
426 
487 
548 
609 


66943 

881 
820 
758 
697 


973 
976 
978 
981 
984 


027 
024 
022 
019 
016 
Oi3 
Oil 
008 
005 
002 


55 

54 
53 
52 
51 
50 
49 
48 
47 
46 


5 

6 
7 
8 
9 

To 

11 

12 
13 
14 


6 
7 
8 
9 
10 
12 
13 
14 
15 


5 
6 

7 
8 
9 
10 
11 
12 
13 
14 
15 
17 
18 
19 
20 


5 

6 

7 
8 

g 

"To 
11 

12 
13 
14 
15 
16 
17 
18 
19 



6 
7 
8 
9 
10 
11 
12 
13 
14 
14 
15 
16 
17 
18 


5 

6 
7 
8 
9 
10 
10 
11 

12 
13 
14 
15 
16 
17 
18 


5 
6 
7 
7 

8 
9 
10 
11 
12 
13 


5 
6 
6 

7 
8 
~9 
10 
11 
12 
13 
14 
15 
16 
16 
17 







~~6 















10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


378 
437 
495 
553 
612 


635 
574 
513 
452 
391 
330 
269 
208 
147 
087 


987 
989 
992 
995 
C98 


670 
728 
786 
844 
902 


670 
731 
792 
853 
913 


01000 
003 
006 
009 
Oil 


000 
98997 
994 
991 

989 


45 

44 
43 
42 
41 




15 

16 
17 
18 
19 


16 
17 
18 
19 
20 


15 
16 
17 
18 
19 


14 
15 
16 
17 
18 


20 

21 
22 
23 
24 
25 
20 
27 
28 
29 


960 
33018 
075 
133 
190 


040 
66982 
925 
867 
810 
752 
695 
638 
580 
523 


974 
34034 
095 
155 
215 


026 
65966 
905 
845 
785 
~724 
664 
604 
544 
484 


014 
017 
020 
022 
025 
"028 
031 
033 
036 
039 


986 
983 
980 
978 
975 
972 
969 
967 
964 
961 


40 

39 
38 
37 
36 
35 
34 
33 
32 
31 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


21 
22 
23 
24 
25 
"26 
27 
28 
29 
30 


21 
22 
23 
24 
25 
~26 
27 
28 
29 
30 


20 
21 
22 
23 
24 
~25 
26 
27 
28 
29 


20 
21 
22 
23 
24 


20 
21 
22 
23 
24 
~25 
26 
27 
28 
29 


19 
20 
21 
22 
23 
~24 
25 
26 
27 
28 


19 
20 
21 
22 
23 
24 
25 
26 
27 
28 


19 
20 
21 

21 
22 
23 
24 
25 
26 
27 


18 
19 
20 
21 
22 
23 
24 
25 
26 
27 


- 





248 
305 
362 
420 
477 
33534 
591 
647 
704 
761 


276 
336 
396 
456 
516 


25 
26 
27 

28 
29 


30 

31 
32 
33 
34 


66466 
409 
353 
296 
239 


34576 
635 
695 
755 
814 


65424 
365 
305 
245 

186 


01042 
045 
047 
050 
053 


98958 
955 
953 
950 
947 


30 

29 
28 
27 
26 




30 

31 
32 
33 
34 


32 

33 
34 
35 
36 


31 
32 
33 
34 
35 


30 

32 
33 
34 
35 


30 
31 
32 
33 
34 


30 
30 

31 
32 
33 


29 
30 
31 
32 
33 


28 
29 
30 
31 
32 


28 
29 
30 
31 
32 


28 
28 
29 
30 
31 


2 
2 
2 
2 
2 




35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


818 
874 
931 
987 
34043 


182 
126 
069 
013 
65957 


874 
933 
992 
35051 
111 


126 
067 
008 
64949 
889 
830 
771 
712 
653 
595 


056 
059 
062 
064 
067 
" 070 
073 
076 
079 
081 


944 
941 
938 
936 
933 


25 

24 
23 
22 
21 
20 
19 
18 
17 
16 


35 

36 
37 
38 
39 


37 
38 
39 
40 
41 


36 
37 
38 
39 
40 


36 
37 
38 
39 
40 


35 

36 
37 
38 
39 


34 
35 
36 
37 

38 


34 
35 
36 
37 
38 


33 
34 
35 
36 
37 


33 
34 
35 
35 

36 


32 
33 
34 
35 
36 


2 
2 
2 
2 
2 


- 


100 
156 
212 
268 
324 


900 
844 
788 
732 
676 


170 
229 
288 
347 
405 


930 
927 
924 
921 
919 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


42 
43 
44 
45 
46 


41 
42 
43 
44 
45 


41 
42 
43 
44 
45 


40 
41 
42 
43 
44 
^45 
46 
47 
48 
49 


39 
40 
41 
42 
43 


39 
40 
41 
42 
43 


38 
39 
40 
41 
42 


37 
38 
39 
40 
41 
42 
43 
44 
45 
46 


37 
38 
38 
39 

40 


2 

2 
2 
2 
2 


45 

46 
47 
48 
49 


380 
436 
491 
547 
602 


620 
564 
509 
453 
398 


464 
523 
581 
640 
698 


536 
477 
419 
360 
302 


084 
087 
090 
093 
096 


916 
913 
910 
907 
904 


15 

14 
13 
12 
11 




47 
48 
49 
50 
51 


47 

48 
49 
50 
51 


46 
47 
48 
49 
50 


44 
45 
46 
47 

48 


44 
44 

45 
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47 


43 
44 
45 
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47 


41 
42 
43 
44 
45 


2 
2 
2 
2 
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2 
2 
2 
2 
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50 

51 
52 
53 

54 


658 
713 
769 
824 
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154 


342 
287 
231 
176 
121 


757 
815 
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243 

185 
127 
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099 
102 
104 
107 
110 


901 
898 
896 
893 
890 


10 

9 

8 
7 
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50 

51 
52 
53 
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52 

54 
55 
56 
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52 
53 
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51 
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50 
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49 
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48 
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48 
48 
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47 
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46 
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2 
3 
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2 
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55 

56 
57 
58 
59 


066 
Oil 
64956 
901 
846 


36047 
105 
163 
221 

279 


63953 
895 
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721 


113 
116 
119 
122 
125 


887 
884 
881 
878 
875 


5 

4 

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55 

56 
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58 
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58 
59 
60 
61 
62 


57 

58 
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60 
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56 
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60 


55 

56 
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54 
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58 


53 
54 
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52 
53 
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56 


51 
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50 

51 
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54 


3 
3 
3 
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2 
2 
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60 


35209 


64791 


36336 


63664 


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60 

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63 


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57 


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54 


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3 




2 
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35209 
263 
318 
373 
427 


54 
55 
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54 
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52 
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52 
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52 
52 
52 

52 
52 
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51 
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51 
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64791 
737 
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627 
573 


36336 
394 
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58 
58 
57 
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57 
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57 
50 
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54 

"d 

1' 


63664 
606 
548 
491 
434 


01128 
131 
133 
136 
139 


3 
2 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
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98872 
869 
867 
864 
861 


60 

59 
58 
57 
56 






1 

2 
3 



1 
2 
3 



1 
2 

^ 




1 
2 
3 




1 
2 
3 



I 
2 
3 




1 

2 
3 



1 
2 
3 
3 



1 
2 
3 
3 
















5 

(5 
7 
8 
9 
10 
11 
12 
13 
14 


481 
536 
590 
644 

698 


519 
464 
410 
356 
302 
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194 
140 
086 
032 


624 
681 
738 
795 
852 


376 
319 
262 
205 
148 


142 
145 
148 
151 
154 


858 
855 
852 
849 
846 


55 

54 
53 
52 
51 




5 

6 
7 
8 
9 


5 
6 
7 
8 
9 


5 

6 
7 
8 
9 


5 
6 
7 
7 

8 


5 
6 
6 

7 
8 


4 
5 

6 

7 
8 


4 

5 
6 

7 
8 


4 
5 
6 

7 
8 


4 
5 
6 

7 
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752 

806 
860 
914 
968 


909 
966 
37023 
080 
137 


091 
034 

63977 
920 
863 


157 
160 
163 
166 
169 


843 
840 
837 
834 
831 
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825 
822 
819 
816 
813 
810 
807 
804 
801 
"798 
795 
792 
789 
786 


50 

49 
48 
47 
46 


10 

11 
12 
13 
14 


10 
11 
12 
13 
14 


10 
10 

11 
12 
13 


9 
10 
11 
12 
13 


9 
10 
11 
12 
13 


9 
10 
11 
12 
13 


9 
10 
11 

11 
12 


9 
10 
10 

11 
12 


8 
9 
10 
11 
12 






1 

1 
1 
1 









15 

16 
17 
18 
19 
30 
21 
22 
23 
24 


36022 
075 
129 
182 
236 
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342 
395 
449 
502 


63978 
925 
871 
818 
764 


193 
250 
306 
363 
419 


807 
750 
694 
637 
581 


172 
175 
178 
181 
184 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 


15 

16 
17 
18 
19 


14 
15 
IB 
17 
18 
19 
20 
21 
22 
23 


14 
15 
16 
17 
18 
19 
20 
21 
22 
23 


14 
15 
16 
17 
18 


14 
15 
16 

16 
17 

~18 
19 
20 
21 
22 


14 
14 

15 
16 
17 
18 
19 
20 
21 
33 


13 
14 
15 
16 
17 
"18 
19 
19 
20 
21 


13 
14 
15 
16 

16 

n 

18 
19 
20 
21 


13 
14 

14 
15 
16 
"17 
18 
19 
20 
30 


1 

2 

2 


1 
1 
1 
1 
1 
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1 
1 
1 
1 




1 
1 

1 

1 
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1 

1 
1 
1 


711 
(558 
605 
551 
498 


476 
532 
588 
644 
700 


524 

468 
412 
356 
300 


187 
190 
193 
196 
199 


30 

21 
22 
23 
24 
35 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


19 
20 
31 

21 
22 


35 

20 
27 
28 
29 


555 

608 
660 
713 
766 
36819 
871 
924 
976 
37028 


445 
392 
340 
287 
234 


756 
812 
868 
924 
980 


244 
188 
132 
076 
020 


202 
205 
208 
211 
214 


35 

34 
33 
32 
31 
30 
29 
28 
27 
26 
35 
24 
23 
22 
21 


24 
25 
26 
27 
28 


24 
25 
26 
27 
38 
28 
29 
30 
31 
32 


23 
24 
25 

26 
27 


23 
24 
25 
26 
27 
28 
38 
29 
30 
31 
32 
33 
34 
35 
36 


22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
33 
34 
35 


22 
23 
24 
25 
36 


22 
23 
33 
24 
25 
26 
27 
28 
39 
29 
^30 
31 
32 
33 
34 
"35 
36 
36 
37 
38 


21 
22 
23 
24 
25 
26 
38 
27 
28 
29 


2 
2 
2 
2 
2 
3 
2 
2 
2 
2 


1 
1 
1 
1 

1 

2 
2 
2 
2 
2 
'2 
2 
2 
2 
2 


1 
1 
1 

1 
1 
1 
1 
1 
1 


30 

31 
32 
33 
34 


63181 
129 
076 
024 
65*972 


38035 
091 
147 
202 
257 


61965 
909 
853 
798 
743 


01217 

220 
223 

226 
229 


98783 
780 
777 
774 
771 


29 
30 
31 
32 
33 
34 
35 
36 
37 
38 


28 
29 
30 
31 
32 


26 
27 
28 
29 
30 
~31 
32 
33 
34 
34 
35 
36 
37 
38 
39 


35 

36 
37 

38 
39 


081 
133 
185 
237 
289 


919 
867 
815 
763 
711 


313 
368 
423 
479 
534 


687 
632 
577 
521 
466 


232 
235 

238 
241 
244 


3 
3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
3 
3 
3 
4 


768 
765 
762 
759 
756 


33 
34 
35 
36 
37 


33 
34 
35 
35 

36 


30 
31 

31 
32 
33 


2 
2 
2 
3 
3 


1 
1 
1 
1 

1 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


341 
393 
445 
497 
549 


659 
607 
555 
503 
451 


589 
644 
699 
754 
808 


411 
356 
301 
246 
192 


247 
250 
254 
257 
260 


753 
750 
746 
743 
740 


30 

19 
18 
17 
16 


40 

41 
42 
43 
44 


39 
40 
41 
42 
43 


38 
39 
40 
41 
42 


37 
38 
39 
40 
41 


37 
38 

38 
39 
40 


36 
37 
38 
39 
40 


34 
35 
36 
37 
37 


3 
3 
3 
3 
3 


3 

2 
2 
2 
2 


1 
1 
1 
1 
1 


600 
652 
703 
755 
806 


400 
348 
297 
245 
194 


863 
918 
972 
39027 
082 


137 
082 
028 
60973 
918 


263 
266 
269 
272 
275 


737 
734 
731 
728 
725 


15 

14 
13 
12 
11 
10 
9 


6 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


44 
44 

45 
46 
47 

Is 

49 
50 
51 
52 


43 
44 
45 
46 
47 


42 
43 
44 
45 
46 


41 
42 
43 
44 
45 
4b 
47 
48 
49 
5( 


40 
41 
42 
43 
44 


40 
41 
42 
43 

43 
44 
45 
46 
47 
48 
49 
49 
50 
51 
52 


39 
40 
41 
43 

42 
43 
44 
45 
4C 
47 


38 
39 
40 
41 
43 
42 
43 
44 
45 
46 


3 

3 
3 
3 
3 
3 
3 
g 

4 
4 


' 2 
2 
2 
2 
2 




50 

51 
52 
53 
54 


858 
909 
960 
38011 
062 


142 
091 
040 
61989 
938 


136 
190 
245 
299 
353 


864 
810 
755 
701 
647 


278 
281 
285 
288 
291 


722 

719 
715 
712 
709 


48 
48 
49 
50 
61 


47 
48 
49 
49 
50 


45 
46 
47 
48 
49 


2 
3 
3 
3 
3 


-\ 


55 

56 
57 
58 
59 


113 
164 
215 
266 
317 


887 
836 
785 
734 
683 


407 
461 
515 
569 
623 


593 
539 
485 
431 
377 


294 
297 
300 
303 
306 


706 
703 
700 
697 
694 


5 

1 


55 

56 
57 
58 
59 


53 
54 
55 
56 
57 
58 


52 
53 
54 
55 
56 
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51 
52 
53 
54 
55 


50 

51 
52 
53 
54 


50 
50 

51 
52 
53 


48 
49 
49 
50 
51 


47 
48 
48 
49 
50 


4 
4 
4 
4 
4 
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2 
3 
3 
3 
3 
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3 


60 


38368 


61632 


39677 


60323 


01310 


98690 





60 


56 


55 


54 


53 


52 


51 




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10. 

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9. 

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10. 

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10. 

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58 


57 


56 


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54 

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53 

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53 

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103 C 



76 

58 



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TABLE II 



165 C 





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51 

50 
51 
50 
50 
51 
50 
50 
50 
50 
50 
50 
50 
50 
40 
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50 
49 
30 
49 
49 
60 
49 
49 
49 
49 
49 
49 
49 

: 

48 
49 
48 
49 
48 
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48 
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48 
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48 
48 
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53 
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54 
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53 

53 
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52 
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53 
52 
53 
52 
63 
52 
52 

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52 

52 
52 
52 
52 
52 
51 
52 
52 
51 
52 
51 
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52 
51 
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51 
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51 
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50 


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10. 


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3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
3 
4 
3 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 
3 
4 
3 
3 

: 

3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
3 
4 
3 
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d" 

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1 COS 

9. 

B8690 
687 
684 
681 
678 
675 
671 
668 
665 
662 
659 
656 
652 
649 
646 
643 
640 
636 
633 
630 
627 
623 
620 
617 
614 
610 
607 
604 
601 
597 
98594 
591 
588 
584 
58 




// 


54 


53 


Pr 
52 


1 
2 
3 
3 


opor 

~0 
1 
2 
3 
3 


tion 
50 


1 
2 
2 
3 


al F 
49 


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48 


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47 


4 

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i 



1 
1 
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i 
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i 
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1 
1 
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1 
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1 
1 
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i 
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2 

2 

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1 

2 
3 
4 


8368 
418 
469 
519 
570 


61632 
582 
531 
481 
430 


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269 
215 

162 
108 


01310 
313 
316 
319 
322 




g 

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r 

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51 
50 
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1 
2 
3 
4 
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6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 



1 
2 
3 
4 



1 
2 
3 
4 



1 

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2| 
3 



1 
2 
2 
3 



1 
2 
2 
3 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
6 
7 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
2V 
30 
31 
32 
33 
34 


620 
670 
721 
771 
821 
871 
921 
971 
39021 
071 
121 
170 
220 
270 
319 
3~69 
418 
467 
517 
566 


380 
330 
279 
229 
179 
129 
079 
029 
60979 
929 
879 
830 
780 
730 
681 
631 
582 
533 
483 
434 
385 
336 
287 
238 
189 
60140 
091 
042 
59994 
945 
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751 

70: 


945 
999 
40052 
106 
159 
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266 
319 
372 
425 
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531 
584 
636 
689 
742 
795 
847 
900 
952 


055 
001 
59948 
894 
841 
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734 
681 
628 
575 
522 
469 
416 
364 
311 
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205 
153 
100 
048 


325 

329 
332 
335 
338 
"341 
344 
348 
351 
354 
357 
360 
364 
367 
370 
373 
377 
380 
383 
386 
390 
393 
396 
399 
403 
01406 
409 
412 
416 
419 
422 
426 
429 
432 
435 


4 
5 
6 


4 

5 
6 


5 

6 


5| 



5 

G 


5 : 5 

6 6 


5 

5 
6 
7 
8 
9 
9 
10 
11 
12 
13 
13 
14 
15 
16 
16 
17 
18 
19 


8 
T 
10 
11 
12 
13 
14 
14 
15 
1G 
17 
18 
19 
20 
21 
22 
22 
23 
24 
25 
26 
27~ 
28 
29 
30 
31 
32 
32 
33 
34 
35 


8 
9 
10 
11 

11 
12 
13 
14 
15 
16 
17 
18 
19 
19 
20 
21 
22 
23 
24 
25 
26 
26 
27 
28 
29 
30 
"31 
32 
33 
34 
34 


8 
~9 
10 
10 
11 
12 
13 
14 
15 
16 
16 
17 
18 
19 
20 
21 
22" 
23 
23 
24 
25 
26 
27 
28 
29 
29 
30 
31 
32 
33 
34 


8 
~8 
9 
10 
11 
12 
13 
14 
14 
15 
16 

17 
18 
19 
20 
20 
21 
22 
23 
24 
25 
26 
26 
27 
28 
29 
30 
31 
31 
32 
33 


8 
8 
9 
10 
11 
12 
12" 
13 
14 
15 
16 
17" 
18 
18 
19 
20 
21 
22 
22 
23 
24 
25 
26 
27 
28 
28 


7j 
8i 
9 
10 
11 
11 

"12" 
13 
14 
15 
1C 
16 
17 
18 
19 
20 
20" 
21 
22 
23 
24 
24 
25 
26 
27 
28 


7 
8 
9 
10 

10 

11 
12" 

13 
14 
14 

15 
16 
17 
18 
IS 
19 
20 
21 
22 
22 
23 
24 
25 
26 
26 
27 


615 
664 
713 
762 
811 
39860 
909 
958 
40006 
055 


41005 
057 
109 
161 
214 
41266 
318 
370 
422 
474 


58995 
943 
891 
839 
786 
58734 
682 
630 
578 
52( 


35 

34 
33 
32 
31 
30 
2 1 
2i 
2' 
26 


20 
20 

21 
22 
23 

24 

25 
26 
27 

27 
28 
29 
30 
31 


2 
2 
2 
2 
2 
2 

2 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
4(> 
47 
48 
49 
5< 
5 
5' 
5< 
> 
55 
5b 
57 
58 
59 


103 
152 
200 
249 
297 


526 
578 
629 
681 
733 
784 
836 
887 
939 
990 
42041 
093 
144 
195 
246 


474 
422 
371 
319 
267 


578 
574 
571 
568 
565 


2 
2< 
2! 
22 
21 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


29 
30 
31 
32 
33 


29 i 28 
29 | 29 
30 30 
31 30 
32 31 


2 
2 
3 
3 


346 
394 
442 
490 
538 
686 
634 
682 
730 
778 


654 
(KM 
558 
510 
462 
414 
I 366 
1 318 
270 
222 


21< 
164 
113 
061 
010 
5795t 
907 
856 
805 
754 


439 
442 
445 
449 
452 
455 
459 
462 
465 
469 


561 

558 
i 55. 

: 55 

548 
515 
54 
538 
535 
53 


20 

19 

1 
1 


36 
37 
38 
39 
40 
40 
41 
42 
43 
44 


35 
36 
37 
38 
39 
40~ 
41 
42 
42 
43 


35 
36 
36 
37 
38 
39 
40 
41 
42 
42_ 
43 
44 
45 
46 
47 
"48~ 
49 
49 
50 
51 
52 
52 
Pi 


34 
35 
36 
37 
37 
38 
39 
40 
41 
42 
42 
43 
44 
45 
46 
47 
48 
48 
49 
50 


33 

31 
35 
36 
37 
38 
38 
39 
40 
41 


33 

33 
34 
35 
3(5 
37 
3S 
38 
39 
40 


32 
33 
34 

34 
35 

36 
37 

38 
38 
39 


31 

32 
33 
34 
34 
35 
36 
37 
38 
38 
39 
40 
41 
42 
42 
43 
44 
45 
45 
46 
47 


3 

! 3 

; 3 

3 
3 

"T 

3 
3 
3 

! 3 


2 

'2 

\ 1 
! 2 

Q 


i 

i 


825 
873 
921 
968 
4101(] 


175 
127 
079 
032 
58984 


297 
348 
399 
450 
501 
552 
603 
653 
704 
755 


70; 

652 
601 
i 55( 

| 499 


472 

475 
479 
482 
485 


528 
525 
52 
51 
51 


50 

51 
52 
53 
54 


45 
46 
47 
48 
49 
^0~ 
50 
51 
52 
53 
54 
54 


44 
45 
46 
47 
48 
19 
49 
50 
51 
52 
53 
53 


42 

42 
43 
44 
45 
46 
47 
48 
48 
49 


41 
42 

42 
43 
44 


40 
41 
42 

42 
43 
44 
45 
46 
46 
47 
~48 

48 

Part 


! 3 
i 3 
3 
4 
4 

"T 
4 
4 
4 
4 


063 
111 
158 
205 
252 


937 

889 
842 
795 

748 


448 
397 
347 
29( 
245 
57195 


489 
492 
495 
499 
502 
01506 


51 
50 
50 
50 
498 


55 

56 
57 
58 
59 
W> 


45 
46 
47 
47 
48 
49~ 
49 
iall 


60 


41300 


58700 


42805 


98494 




51 
5f 

ropo 


50 
50 

rtioi 


/ 


9. 

/cos 


d 

r 


10. 

Zsec 


9. 

Zcot 


d 
1' 


10. 

Han 


10. 

/ CSC 


9. 

/ sin 


/ 




47 

s 


4 


3 


104 75 

59 





15 C 



TABLE II 



164 C 



' 


/sin 
9. 


d 

l 1 


Jcsc 
10. 


/tan 
9. 


d 
1' 


/cot 
10. 


1 sec 
10. 


d 

i / 


/COS 

9. 


> 




// 


51 


50 


b 

49 


ropo 
48 


rtioi 
47 


aal 
46 


Part 
45 


s 
44 


4 


3 




1 

2 
3 

4 


41300 
347 
394 
441 

488 


47 
47 
47 
47 
47 


58700 
653 
606 
559 
512 


42805 
856 
906 
957 
43007 


51 
50 
51 
50 
50 
il 
50 
50 
50 
50 
50 
50 
50 
50 
50 
49 
50 
50 
49 
50 
49 
50 
49 
50 
49 
49 
49 
50 
49 
49 
49 
49 
49 
49 
49 
48 
4i 
49 
48 
49 
49 
48 
49 
48 
48 
49 
48 
48 
48 
49 
48 
48 
48 
48 
48 
48 
47 
48 
48 
48 


57195 
144 
094 
043 
56993 


01506 
509 
512 
516 
519 


3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
4 
3 
4 


98494 
491 
488 
484 
481 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 




1 
2 
3 
4 
5 
6 
7 
8 
9 



1 
2 
3 
3 




1 

2 

2 
3 



1 
2 

2 
3 



1 
2 
2 
3 



1 
2 
2 

3 



1 
2 
2 

3 



1 
2 
2 

3 



1 
1 
2 
3 
















5 

6 

7 
8 
9 


535 
582 
628 
675 
722 


47 
46 
47 
47 
46 
47 
46 
47 
46 
47 
46 
46 
47 
46 
46 
46 
46 
46 
46 
45 
46 
46 
46 
45 
46 
45 
46 
45 
46 
45 
45 
46 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
45 
44 
45 
44 
45 
44 
45 
44 
44 
44 
45 
44 
44 


465 
418 
372 
325 
278 


057 
108 
158 
208 
258 


943 
892 
842 
792 
742 


523 
526 
529 
533 
536 


477 
474 
471 
467 
464 


5 
6 
7 
8 
8 
9 
10 
11 
12 


5 
6 
7 
8 
8 
9 
10 
11 
12 


5 
6 

7 


5 
6 
6 


5 

5 
6 


5 
5 

6 


4 
5 
6 


4 
5 
6 






1 
1 
1 
1 









10 

11 
12 
13 

14 


768 
815 
861 
908 
954 


232 
185 
139 
092 
046 


308 
358 
408 

458 
508 


692 
642 
592 
542 
492 


540 
543 
547 
550 
553 


460 
457 
453 
450 
447 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


8 
9 
10 
11 

11 


8 
9 
10 

10 

11 


8 
9 

9 
10 
11 


8 
8 
9 
10 
11 


8 
8 
9 
10 

10 


7 

8 
9 
10 
10 




1 
1 
1 
1 


15 

16 
17 
18 
19 


42001 
047 
093 
140 
186 


57999 
953 
907 
860 
814 


558 
607 
657 
707 
756 


442 
393 
343 
293 
244 


557 
560 
564 

567 
571 


443 
440 
436 
433 
429 


45 

44 
43 
42 
41 


13 
14 

14 
15 
16 


12 
13 
14 
15 
16 


12 
13 
14 
15 
16 
16 
17 
18 
19 
20 
20 
21 
22 
23 
24 


12 
13 
14 
14 
15 


12 
13 
13 

14 
15 


12 
12 

13 
14 
15 
15 
16 
17 
18 
18 
19" 
20 
21 
21 
22 


11 
12 
13 
14 
14 


11 
12 

12 
13 
14 
15 

15 
16 
17 
18 




1 
1 

1 
1 
1 


20 

21 
22 
23 
24 


232 
278 
324 
370 
416 


768 
722 
676 
630 
584 


806 
855 
905 
954 
44004 


194 
145 
093 
046 
55996 


574 

578 
581 
585 
588 


426 
422 
419 
415 
412 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 


17 
18 
19 
20 
20 


17 
18 
18 
19 
20 
2f 
22 
22 
23 
24 


16 
17 
18 

18 
19 


16 

10 
17 
18 
19 


15 
16 

16 
17 
18 


2 
2 


1 

1 
1 
1 
1 


25 

26 
27 
28 
29 


461 
507 
553 
599 
644 


539 
493 
447 
401 
356 
57310 
265 
219 
174 
128 


053 
102 
151 
201 
250 


947 
898 
849 
799 
750 


591 
595 
598 
602 
605 


409 
405 
402 
398 
395 


35 

34 
33 
32 

31 


25 

26 

27 
28 
29 


21 
22 
23 
24 
25 


20 
21 
22 
22 
23 


20 
20 

21 
22 
23 


19 
20 
20 
21 
22 


18 
19 
20 
21 
21 
22" 
23 
23 
24 
25 
26 
26 
27 
28 
29 
29 
30 
31 
32 
32 


2 
2 
2 
2 

2 


1 
1 
1 

1 
1 


30 

31 
32 
33 
34 


42690 
735 
781 
826 
872 


44299 
348 
397 
446 
495 


55701 
652 
603 
554 
505 


01609 
612 
615 
619 
623 


98391 
388 
384 
381 
377 


30 

2< 

28 
27 
26 


30 

31 
32 
33 
34 


26 
26 

27 
28 
29 


25 

26 
27 
28 
28 


24 
25 
26 
27 
28 


24 
25 
26 

26 
27 


24 
24 
25 
26 
27 


23 
24 
25 
25 

26 


22 
23 
24 
25 
26 
26 
27 
28 
28 
29 


2 

2 
2 
2 
2 


2 
2 
2 

2 
2 
2 
2 
2 
2 
2 
2" 
2 
2 
2 
2 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


917 
962 
43008 
053 
098 


083 
038 
56992 
947 
902 


544 
592 
641 
690 
738 


456 
408 
359 
310 
262 


627 
630 
634 
637 
641 


373 
370 
36C 
363 
359 


25 

24 
23 
22 
21 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


30 
31 

31 
32 
33 
34 
35 
30 
37 
37 


29 
30 
31 
32 
33 


29 
29 
30 
31 
32 
33 
33 
34 
35 
36 


28 
29 
30 
30 
31 


27 
28 
29 
30 
31 
31 
32 
33 
34 
34 


27 
28 
28 
29 
30 


2 
2 
2 
3 
3 


143 
188 
233 
278 
323 


857 
812 
767 
722 
677 


787 
836 
884 
933 
981 


213 
164 
116 
067 
019 


644 
648 
651 
655 
658 


356 
352 
349 
345 
342 


2( 


33 

34 
35 
36 
37 


32 
33 
34 

34 
35 


31 

31 
32 
33 
34 


30 
31 
32 
32 

33 


3 
3 
3 
3 
3 


45 

46 
47 
48 
49 


367 
412 
457 
502 
546 


633 
588 
543 
498 
454 


45029 
078 
126 
174 
222 


54971 
922 

874 
826 
778 


662 
666 
669 
673 
676 


338 
334 
331 
327 
324 


15 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


38 
39 
40 
41 
42 


38 
38 

39 
40 
41 


37 
38 
38 
39 
40 


36 
37 
38 
38 
39 


35 
36 
37 
38 
38 


34 
35 
36 
37 
38 


34 

34 
35 
36 
37 


33 
34 

34 
35 
36 


3 

3 
3 
3 
3 


2 
2 

2 
2 
2 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


591 
635 
680 
724 
769 


409 
365 
320 
276 
231 


271 
319 
367 
415 
463 


729 
681 
633 
585 
537 


680 
683 
687 
691 
694 


320 
317 
313 
309 
306 


10 


42 
43 
44 
45 
46 


42 

42 
43 
44 
45 


41 
42 

42 
43 
44 


40 
41 
42 

42 
43 


39 
40 
41 
42 
42 
~43 
44 
45 
45 
46 


38 

39 
40 
41 
41 
42 
43 
44 
44 
45 


38 
38 
39 
40 
40 
41 
42 
43 
44 
44 
45 
"45 
Par 


37 

37 
38 
39 
40 
40 
41 
42 
43 
43 


3 
3 
3 
4 
4 
~T 
4 
4 
4 
4 


2 
3 
3 
3 
3 
3 
3 
3 
3 
3 


813 
857 
901 
946 
990 


187 
143 
099 
054 
010 


511 
559 
606 
654 
702 


489 
441 
394 
346 
298 


698 
701 
705 
709 
712 


302 
299 
295 
291 

288 


47 
48 
48 
49 
50 


46 
47 
48 
48 
49 


45 
46 
47 
47 

48 


44 
45 
46 
46 

47 


44034 


55966 


45750 


54250 


01716 


98284 


I 


60 


51 


50 


49 


48 


47 


46 


44 


4 

~T 


3 
~3 


/ 


9. 

/cos 


d 
1' 


10. 

/sec 


9. 

/cot 


d 
1 


10. 

/tan 


10. 

/ CSC 


d 

1' 


9. 

/ sin 


/ 




51 


50 


49 

P 


48 
rope 


47 

>rtio 


46 

nal 


44 

ts 


105 


74 

60 







16 C 



TABLE II 



163 C 



, I Kill d 
9. i' 


I CHC 

10. 


tan d 
9. 1' 


Zcotl 

JM 

54250C 

155 
108 
060 


1 see 
10. 
H716 
719 
723 
727 
730 


1 COS , 

9. 




1 
2 
3 

4 


8 

1 
2 

2 
3 


7i 


16 


15 


tion 


"o 


arts 
2 




1 


4 


3 

6 


044034 
1 078** 
2 122** 
3 166** 
4 210** 


55966 
922 
878 
834 
790 


5750,, 
797*! 
845*? 
892*; 
940** 


98284 60 



1 
2 
2 
3 



1 
2 
2 
3 











281 59 
27758 
27357 
27056 


2 
2 

3 


1 
2 
3 


1 
2 
3 


1 
2 
3 


1 

2 
3 






o" 










5 253 4 . 

6 297** 
7 341 ** 
8 385** 
9 428 *'J 


747 
703 
659 
615 
572 


987., 
6035*, 

noo '' 


013 
53965 
918 
870 
823 
776 
729 
681 
634 
587 


734 

738 
741 
745 
749 


26655 
26254 


5 

6 


4 
5 


4 
5 


4 
5 


4 

4 


4 

4 


4 
4 


4 
4 


3 
4 








i 
i 
i 
i 


082 
130* 
177* 


259 53 
25552 
25151 


8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


6 


6 


6 


6 


6 


6 

6 


6 
6 


5 
6 


10 472 
11 516*, 
12 559*, 
13 602*; 
14 646*; 


528 
484 
441 
398 
354 


224 ; 

271* 
319*! 
306* 
413*. 


752 
756 
760 
763 
767 


24850 
24449 
24048 
23747 
23346 


8 
9 




1 


8 
9 

9 


1 


8 
8 
9 
10 
1 


8 
8 
9 





7 

8 
9 




7 
8 
9 
9 



7 
8 
8 
9 
10 


7 
8 
8 
9 



15 689 ^ 
16 733*! 
17 776 * l 
18 819 * l 
19 862*' 


311 
267 
224 
181 
138 


460. 
507*. 
554* 
601* 
648* 


540 
493 
446 
399 
352 


771 
774 

778 
782 
785 


22945 
22644 
22243 

21842 
21541 


2 
3 
4 
14 

5 
6 
17 
18 
18 
19 


2 
13 
13 

14 
15 
16 

16 
17 
18 
19 


12 
12 

13 
14 
15 


1 

2 
3 
4 
14 

15 
16 

16 
17 
18 


11 
12 

12 
13 
14 
15 

15 
16 
17 
18 


1 

1 
2 
3 
4 
14 
15 
16 
16 
17 


10 

11 

12 
13 
13 

14 
15 

15 
16 

17 




1 
2 
12 

3 


"T 

i 
i 

2 

2 


i 
i 

i 

i 
i 
l 

i 
i 

i 
i 


20 905 . 

21 948* 
22 992* 
2345035* 
24 077 * 


095 
052 
008 
54965 
923 


694. 

741 1 

788* 

835* 
881* 


306 
259 
212 
165 
119 


789 
793 
796 
800 
804 


21140 
20739 
20438 
20037 
19636 


15 

16 
17 
18 
18 


14 

14 
15 
16 

16 


25 120, 
26 163 * 4 
27 206* 
28 249 4 
29 292 \ 


880 
837 

?st 

J 7?)S 
.54666 
j 623 

: ss 

2 53 
' 49 


928 
975* 
470211* 
068* 
114i* 
47160 
207 
253 
299 
346 


072 
025 

52979 
932 

886 


808 
811 
815 
819 
823 


19235 

ill 


25 

26 
27 
28 
2( 

30 

31 
32 
33 
34 


20 
21 
22 
22 

23 
24 
25 
26 

26 
27 

28" 
29 
30 
30 
31 


20 
20 
21 
22 
23 
24 
24 
25 
26 
27 
27 
28 
29 
30 
31 


19 
20 

21 
22 


19 
20 
20 

21 


18 

19 
20 
21 
21 


18 
19 
19 
20 
21 


18 
18 
19 
20 
20 


17 
18 
18 
19 
20 
20 
21 
22 
23 
23 


2 
2 
2 
2 
2 
2 


~ 


3045334. 
31 377* 
32 419 
33 462 
34 504 


52840 
793 
747 
701 
654 


01826 
830 
834 
838 
841 


9817430 

17029 
16628 
162 27 
1592 


22 
M 
25 
25 

26 

28 
28 
29 
30 


2? 
24 
25 
26 
26 
27 
28 
28 
29 
30 
31 
32 
32 
33 


23 
23 
24 
25 
26 
26 
27 
28 
29 


22 

23 
24 

24 


21 
22 
22 
23 
24 


35 547 

36 589 
37 632 
38 674 
39 716 


2 45 
3 4l 

2 36 

2 32 

2 28 


392 
438 
484 
530 
576 


608 
562 
51f 
47( 
424 


845 
849 
853 
856 
860 


1552 
1512 
1472 
1442 
1402 


35 

36 
37 
38 
39 


25 
26 
27 
27 

28 


24 
25 
26 
27 
27 


24 
25 
25 

26 
27 


\ 


40 758 
41 801 
42 843 
43 885 
44 927 


24 

2 19 

2 }* 

2 


622 
668 
714 
760 
806 


378 
332 
28( 
240 
19 


864 
868 
871 
875 
879 


1362 
1321 
1291 
1251 
1211 


40 

41 
42 
43 
44 


32 
33 
34 

34 
35 


31 

32 
33 
34 
34 
35 
36 
37 
38 
38 


31 

31 
32 
33 
34 


29 

30 
31 
32 
32 
33 
34 
34 
35 
3 
3 
3 
3 
3 


29 

29 
30 
31 
32 
32 
33 
34 
34 
35 


28 
29 
29 
30 
3 
32 
3 
33 
34 
3 


27 

28 
29 

29 
30 


~3 




45 969 

4646011 
47 053 
48 095 
49 136 


2 3 

^5398 

Q4. 

., t 

\ 90 
J 86 


852 
897 
943 
989 
48035 


14 
10 
05 
01 
5196 


883 
887 
890 
894 
898 


1171 
1131 
1101 
1061 
1021 


45 

46 
47 
48 
49 


36 
37 
38 
38 
39 
40 
41 
42 
42 


34 
35 
3f 
37 
38 
3 
39 
4 
4 


34 

34 
35 
36 
3 
3 
3 
3 
4 


31 

31 
32 
33 

33 




50 178 
51 220 
52 262 
53 303 
54 345 


2 82 

I 78 
73 
J 69 

l 65 


080 
126 
171 
217 
262 


92 
87 
82 
78 
73 


902 
906 
910 
913 
917 


0981 
094 
090 
087 
083 


50 

5 
5 
5 
5 


39 
40 
41 
42 


3 
3 
3 
3 
3 


3 
3 
3 
3 
3 


34 
35 
36 
36 
37 


X 




55 386 

56 428 
57 469 
58 511 
59 552 


s 

11 53 
12 4* 
U ^ 
12 44 


307 
353 
398 
443 
489 


69 
64 
60 
55 
51 


921 
925 
929 
933 
937 


079 
075 
071 
067 
063 


5 

5 
5 
5 
5 


44 
45 
4 
4 
4 


43 
44 
45 

45 
46 


4 
4 
4 

44 
4 


4 
4 

4 
44 


4 

4 
4 


3 
4 
4 

4 


3 
3 
4 
4 


38 
3 

39 
4 






6046594 


5340 


48534 


5146 


01940 


98060 


6 


4 


47 


4 


4 


44 


4 
4 
mal 


4 
4 
Pai 


4 
4 

is 






, 9. 

1 COS 


d 10. 

l' 1 sec 


9. 

Zcot 


10. 

ttai 


10. 

Ics 


9. 

Ism 


' 


4 


47 


1 


45 

top 


4 

ortic 






106 73 

Cl 



17 



TABLE II 



162 f 



/ 1 Sill 

9. 


d / CSC 

i' 10. 


I tan 
9. 


d I Cot 

i' 10. 


1 sec 
10. 


d /COS , 

1' 9. 


rt 


45 

o" 

1 
2 
2 
3 


44 

"o 
1 

i 

2 
1^ 


1 

JB 


1 

1 


'rop 
42 


DftlC 

41 



1 

1 

2 
3 


mal 
40 


Par 
39 


ts 
5 


4 


3 








46594 
1 635 
2 676 
3 717 

4 758 


41 5340 
: 36 
32 
283 
I 24 


48534 
579 
624 
669 
714 


,r 5146 
K 42 
' 37 

: 33 

S 28 


01940 
944 
948 
952 
956 


A 98060 6fl 
* 0565 
I 052 5 

* 0485 
* 0445 




] 



1 
1 
2 
3 




i 
1 

3 




1 

1 

2 
3 
















5 800 

6 841 
7 882 
8 923 
9 964 


11 20 
15 

11 
07 

1 03 


759 
804 
849 
894 
939 
-984 
49029 
073 
118 
163 


w 24 

5 19C 

16 

5 10 

r 06 

,,~6I 

^5097 

u 2 
5 88 
1 83 


960 
964 
968 
971 
975 


. 0405 
* 0365 
* 0325 
I 0295 
* 025 5 
4 ~02i 5 
* 0174 
* 0134 
* 0094 
* 0054 
4 "00145 
* 97997 4 
* 9934 
* 9894 
I 9864 


5 


4 


4 


i 


4 


3 


3 


3 





1 
1 

1 
1 
1 

1 
1 
i 

i 
i 

2 
2 






1 
1 
1 

1 

1 
1 
1 

1 
1 

1 
1 


(I 







1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
] 
1 
1 
1 
1 
1 


i 
8 
c 

10 

11 

If 
i 

13 
14 


5 
6 
7 
8 
8 
9 
10 
10 

11 

12 
13 
14 
14 
15' 
16 
16 
17 
18 
T9 
20 
20 

21 

22 


5 
6 

7 


5 
6 

r 

8 
9 
9 

10 

11 

11 

12 
13 
14 
14 
15 
16 
16 
17 
18 
19 
19 
20 
21 
22 
23 
23 
34 
24 
25' 
20 
27 
27 
28 
29 
29 
30 
31 
32 
32 
33 
34 
34 
35 
~3b r 
37 
37 
38 
39 


5 
6 
6 
7 
8 
8 
9 
10 
10 
11 

13 
13 


5 

5 
6 

7 
8 
8 
9 

to 

10 

11 
12 
12 

13 


fi 
5 

6 
7 
7 
8 
9 
9 
10 

11 

11 

12 
13 
13 
14 
15 
15 
16 
17 
17 
1'8 
19 
19 
2l7 
21 
21 


5 
5 
6 


1047005 
11 045 
12 086 
13 127 
14 168 


5299 
, 95 

91 

87 

J 83 


979 
983 
987 
991 
995 


7 

8 
9 
10 
10 


6 
7 
H 
8 
9 
10 
10 

11 

12 

12 


15 209 

16 249 
17 290 
18 330 
19 371 


o 79 
? 75 
i 710 
? 670 
o 62 < 


207 
252 
296 
341 
385 


5 79 

& 

\ 85 


999 
02003 
007 

on 

014 


15 

16 
17 
18 
19 


11 
12 

12 
13 
14 
15 

15 
10 
17 

18 

18 
19 
20 
21 
21 


20 411 
21 152 
22 492 
23 533 
24 573 
25 613. 
26 654* 
27 691* 
28 734* 
29 774* 


. 589 

i 548 
J 508 

i 4fi7 
o 427 
387 

i 34 

306 
266 

0_ 22 _ 6 
52186 

2 14( > 
o 106 
o 066 

n 02 


430 
474 
519* 
563 
607 ; 
652! 
696* 
740 \ 
784* 
828* 


^ 570 
1 526 
^ 48 
4 437 
* 393 
' 348 
A 304 
* 260 
* 216 
4 172 


018 
022 
026 
030 
034 
038 
042 
046 
050 
054 


d 9824 
T 9783 
J 974 38 
7 970 3 
; 966 3fa 
. "96235 
J 95834 
95431 
950 32 
I 946 31 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


14 
15 

15 
16 
17 
IK 

18 

19 
20 
20 


14 

14 
15 
16 
16 
17 
18 
18 
19 
20 
20 
21 
22 
23 
23 


13 
14 
14 

15 
16 
16 
17 
IS 
18 
19 
'20 
20 
21 


2 
2 



2 

2 

2 
2 
2 

2 
2 

2 
3 
3 


1 
1 
1 

2 
2 
2 
2 
2 

3 


3047814 
31 854* 
32 894* 
33 934* 

*L_14 
3548014 
36 054* 
37 094* 
38 133 
39 173* 
40 Til* 
41 252 1 
42 292* 
43 332* 
44 371^ 

"45" 411* 
46 450] 
17 490* 
48 529?, 
49 568 
50 " 607 
51 647*J 
52 686 ; ; 
53 725* 
54 764 3 ; 


49872. 
916* 
960* 
50004* 
048* 


.50128 
* 084 
4 040 
* 49996 
* 952 

A 908 

* 864 
J 820 
A 777 
J 733 


02058 
062 
066 

070 ; 

074; 

078, 
082, 
08( 
090 
094 
"""098 
102 
106 
110 

_1 

118 4 

122 
126 
130 

134 
139 
143 
147* 
151 
155 J 


. 97942 30 
93829 
93428 
930 27 
; 926 26 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


22 
23 
24 
25 
20 
26 
27 
28 
28 
29 
30' 
31 
32 
33 
33 
34 
34 
35 
36 
M 
38 
18 
39 
40 
40 


22 
23 
23 
24 
25 
26 
20 
27 
28 
29 
29 
30 
31 
32 
32 
33~ 
34 
34 
35 
36 
37 
37 
38 
39 
40 
40 
41 
42 
43 
43 


21 
22 

22 
23 
24 
24 
25 
26 
27 
27 
28" 
29 
29 
30 
31 
32 
32 
33 
34 
34 
35 
36 
36 
37 
38 


2 
2 
2 


23 

23 
24 
25 
25 
26 
27 
27 
28 
29 
29 
30 
31 
31 
32 
33 
33 
34 
35 
35 
36 
37" 
37 
38 
39 
39 
40 


22 
23 

23 
24 
25 
25 
26 
27 
27 
28 
29 
29 
30 
31 
31 
32 
32 
33 
34 
34 
35 
36 
36 
37 
38 
38 
39~ 


3 
3 
3 

3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
t 
4 
4 
4 
4 
4 
4 
4 
5 
5 
5 
5 
5 
5 


2 

2 
3 
3 


2 
2 

2 
2 


51 946 
J 906 

1 827 
D . Z 

J 787 
' 748 
. 708 
, 668 
} 629 
,~~589 
550 
510 
471 
J 432 
, 393 
353 
314 
' 275 
236 


092. 
136* 
180* 

223; 

267* 


. 922 35 

918 24 
91423 
91022 
| TO 21 


24 
25 
25 
20 
27 
27 
28 
29 
29 
30 
31 
31 
32 
33 
33 
34 
35 
3fi 
30 
37 


311 , 
355* 
398* 
442* 
485* 
529. 
572* 
616* 
659* 
703* 
746 
789* 
833*' 
876*! 

919*: 


A 689 
* 645 
602 
; 558 

!..? 
, 471 
428 
384 
341 
297 
,254 
211 
167 
124 
081 


. 902 20 
89819 
894 18 
890 17 
[ 88616 
, 882 15 
878 14 
1 87413 
870 12 
86611 
861 10 
857 9 
853 8 
849 7 
845 6 


3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
4 


2 

o 

2 
2 
o 

2 
2 
2 
2 
2 
2 
3 
3 
3 
3 
3' 
3 
3 
3 
3 
3 
3 


55 803 ' 

56 842 '1 
57 881 i; 
58 920 J 
59 959 


197 
158 
119 
080 
041 


962 * 

1005*: 

048*: 
092*^ 
135 ' 


. 038 
48995 
1 952 
908 
865 


159 A 
163 * 
167* 
171 * 
176* 


841 5 
837 4 
833 3 
829 2 
825 1 


55 

56 
57 
58 
59 


41 
42 
43 
44 
44 


39 
40 
41 
42 
42 


38 
39 
40 
41 
41 
42" 


38 
38 
39 
40 
40 


6048998 


51002 


1178 


8822 


2179 


97821 


60 

-/r 


45 


44 
44 


43 


41 
41 

>rtio 


4 
4 


/ 9. d 

cos 


10. 

/sec 


9. d 

cot i 


10. 

I tan 


10. d 

CSC 


9. , 

Ism 


45 


43 

P 


42 

tODC 


40 

nal 


39 

Par 


5 

ts 


107 


72 

62 





18 C 



TABLE II 



161 C 



/ t sin d 
9. i 


I CSC 

10. 


tan d 
9. i 


icot 
10. 


sec 
10. 


icos , 
9. 


n 


43 


42 


Pro 


porti 
39 


onal 
38 


Pai 
37 


ts 
36) 5 


4 


48998 or 
149037* 
2 076 H 
3 115, 
4 153^ 


51002 
,50963 
924 
885 
847 


1178 .. 

221*: 

264*; 
306*; 

349*; 


48822 
779 
736 

; 694 

651 


K8179 
183 
188 
192 
196 


97821 60 
817 55 




1 

2 
3 
4 
5 

6 


















1 



1 
1 
2 
















812 = 
80857 
80456 


1 
2 
3 


2 
3 


2 
3 


2 
3 


2 
3 


2 

2 


5 192 

6 231 i 
7 269^ 
8 308'; 
9 347* 


. 808 
769 
731 
692 
653 


392' 

435*; 
478* 
520* 
563* 


. 608 
565 
522 
: 480 
437 


200 
204 


80055 
79654 
79253 
78852 
78451 


4 


4 


3 


3 


3 


3 


* 




1 
1 

1 






1 
1 
T 
i 

i 
i 
i 


208 
212 
216 


8 
9 


6 

6 


6 
6 


5 
6 


5 

6 


5 
6 


5 




5 


10 385 , 
11 424* 
12 462, 
13 500* 
14 539^ 


615 
576 
* 538 
: 500 
461 
"423 
, 385 
R 346 
1 308 
R 270 


606* 

648* 
691* 
734* 
776* 


, 394 

: 352 

? 309 
1 266 
: 224 


221 
225 
229 
233 
237 


77950 

77549 
77148 
76747 
76346 


10 

11 
12 
13 
14 


7 
8 
9 
9 
10 


7 
8 
8 
9 
10 
10 

11 

12 
13 
13 


7 
8 
8 
9 
10 


6 
7 
8 
8 
9 
10 
10 

11 
12 

12 


fi 

7 
8 
8 
9 

10" 
10 
11 

11 
12 


6 
7 

7 
8 
9 
~9 
10 
10 
11 
12 


ft 
7 
7 

8 

S 


1 

1 
1 
1 
1 
1 
1 

2 


15 577 1 

16 615^ 
17 654 * 
18 692:; 
19 730^ 


819* 
861* 
903* 
946* 
988* 
"2031 * 
073* 
US' 
157* 
_20_0 
242 
284* 
326 J 
368* 
410 


, 181 
2 139 
\ 097 
j 054 
\ 012 


241 
246 
250 
254 

258 


75945 

75444 
75043 
74642 
t 74241 


15 

16 
17 
18 
19 
2<T 
21 
22 
23 
24 


11 

11 
12 
13 
14 
14 
15 
16 
10 
17 


10 

11 
12 
12 

13 


9 
10 
10 

11 

11 

"l2~ 
13 
13 
14 

14 

"is" 

1C 
16 
17 

17 

"is" 

19 
19 
20 

20 

21 
22 
22 
23 

23 
24 
25 
25 
26 
20 
27 
28 
28 
29 

29 
30 
31 
31 
32 
32 
33 
34 
34 
35 
35 
36 
36 
arts 


l 
i 
i 
i 
i 


20 768' 
21 806;? 
22 844* 
23 882 * 
24 920^ 


232 

1 194 

1 m 

I ll8 
1 080 


,47969 
i 927 
2 885 
" 843 
o 800 


262 
266 
271 
275 
279 


73840 
1 734 39 
* 72938 
1 725 37 
72136 


14 
15 

15 
Hi 
17 


14 

14 
15 
16 

Ifi 


13 
14 

It 

16 
16 

17 
18 
18 
19 


13 
13 

14 
15 
15 


12 
13 
14 
14 
15 
15" 
10 
17 
17 
18 
18" 
19 
20 
20 
21 
~22~ 
22 
23 
23 
24 
~25 
25 
20 
27 
27 


2 
2 
2 
2 
2 
2 

2 
2 
2 


i 

i 
i 

2 
2 
~2~ 

2 
2 

: ^ 


25 958 _ 
26 996;; 
27 50034 i 
28 072; 
29_110; 


8 42 

1 004 
1 49966 
! 928 
I 890 


J 758 
2 716 
I 074 

2 32 

2 590 


283 
287 
292 
296 
300 


71735 
f 71334 
70833 
70432 
J 70031 


25 

26 
27 
28 
29 


18 
19 
19 
20 
21 
22 
22 
23 
24 
24 
25 
26 
27 
27 
28 
19 
29 
30 
31 
32 


IS 
18 

19 
20 
20 


17 
18 
18 
19 
20 


16 

IG 
17 
18 
18 
"19^ 
20 
20 
21 
22 
22 
23 
23 
24 
25 
25 
26 
27 
27 
28 
'28 
29 
30 
30 
31 


30 501 48 Q 

31 185? 
32 223 ' 
33 261 ? 
34 298 


, 49852 
I 815 

1 777 
* 739 

8 7 2 


52452 A 
494 
536, 
578 
(520 


47548 
1 506 
i 464 
i 422 

; 380 

2^339 
2 297 
2 255 
2 213 
_J71 
2 130 
f 088 
1 047 
o 005 
f 46963 


02304 
309 
313 
317 
321 
326 
330 
334 
338 
343 
~ 347 
351 
355 
360 
364 
368 
372 
377 
381 
385 


p 97696 30 
J 691 29 

68728 
* 683 27 
J 679 2f 


30 

31 
32 
33 
34 


21 
22 
22 
23 
24 


20 
21 
22 
23 
23 
24 
25 
25 
26 
27 
27 
28 
29 
29 
30 


20 
20 
21 

21 
22 
23 
23 
24 
25 
25 
"2fi 
27 
27 
28 
29 


o 
3 
3 
3 
3 


2 

2 
2 
2 

2 


35 336'. 
36 374* 
37 41 T 
38 449 
39 486; 
40 523 , 
41 561 ; 
42 598 : 
43 635: 
44 673; 
45 710: 
46 747: 
47 784; 
48 821 ; 
49 858; 


664 
626 

8 589 
! 551 

514 

8 47? 

8 43< 
402 

8 365 
.5 327 

7 2W 

7 253 

7 216 
J7 179 

j? ;L 

18 142 


661 
703 
745 
787 
829 
870 
912 
953 
995 
53037 


' 67425 
* 670 2 
* 666 2 
* 662 2 
1 6572 
A (553 2 
* 649 1 
* 6451 
\ 6401 
* 6361 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


24 
25 
26 
27 
27 
28 
29 
29 
30 
31 
32 
32 
33 
34 
34 


3 
3 

i 3 
3 
3 
~3 

4 
4 
4 
4 
4 
4 
4 
4 
"4 
4 
4 
4 
4 
-"5 

5 
5 
5 


2 
2 
2 
3 
3 

IT 

; ;j 

I I 
i 

I 
i 

I 4 

i 


078 
120 
161 
202 
244 


2 922 
* 880 

1 83 

2 79 

1 75 


632 1 

* 6281 
? 6231 
* 6191 
* 615 1 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
5f 
57 
58 
59 
60 


32 

33 
34 

34 
35 


31 

31 
32 
33 
33 


29 

30 
31 
31 
32 

32~ 
33 
34 

34 
35 
36 

3 
37 
38 
38 
"39" 


28 
28 
29 
30 
30 
31 
31 
32 
33 
33 
34 
35 
35 
36 
M 
37 
37 
ilP 


50 896 l 
51 933' 
52 970' 
53 51007 
54 043 


1 - 

2 030 

4 T 


285 
327 
368 
409 
450 
492 
533 
574 
615 
656 


2 71 

i ( > 7 
63 

59 
" 55 


390 
394 
398 
403 
407 
411 
416 
420 
424 
429 


\ 6 10 * 
* 606 
4 602 
\ 597 

4 593 


36 
37 
37 
38 
39 
39 
40 
41 
42 
42 
43 


35 
36 

36 
37 
38 
38 
39 
40 
41 
41 


31 
35 
3C 
36 
37 
38 
38 
3V 
4( 
40 


32 
32 
33 
34 
34 
35 
1 35 

i :w 
|37 

i 37 

" 38" 


55 080 

56 117 
57 154 
58 191 
59 227 


s 

7 84 

I 80 

36 77 
17 ' ' 


i 50 

4(i 
42 

38 
34 


589 
'\ 584 
! 580 
* 576 
4 571 


6051264 


7 4873 


53697 


4630 


02433 


97567 


42 


41 


5; 4 


/ cos 


d 10. 

]' /SP( 


9. 

/cot 


d 10 

i /tan 


10. 

/ CSC 


d 9. 

i' /sin 


// 


43 


42 


41 

Pr 


39 38 

oporfion 


5 4 



108 



71 



19 C 



TABLE II 



r 



1 

r 

r 
^ 

5 

( 
t 
8 
9 


I sin 
9. 


d 

1' 


I CSC 

10. 


Han 
9. 


d 
1' 


f cot 
10. 


/ sec 
10. 


d 
1' 


1 COS 

9. 


> 







41 


40 


Pr< 
39 


)pon 
37 


tiona 
36 


UPa 
35 


rts 
34 


5 


4 


51264 
301 
338 
374 
411 


37 
37 
36 
37 
36 
37 
36 
37 
36 
36 
37 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
35 
35 
35 
35 
35 
36 
34 
35 
35 
35 
35 
35 
35 
34 
35 


48736 
699 
662 
626 
589 


53697 
738 
779 
820 
861 


41 
41 
41 
41 
41 
41 
41 
41 
40 
41 
41 
40 
41 
41 
40 
41 
40 
41 
40 
41 
40 
41 
40 
40 
41 
40 
40 
41 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
40 
39 
40 
40 
40 
39 
40 
40 
39 
40 
39 
40 
39 
40 
39 
40 
39 
39 
40 

~d 

1' 


46303 
262 
221 
180 
139 


02433 
437 
442 
446 
450 


4 
5 
4 
4 
5 
4 
5 
4 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
4 
5 
4 
5 
4 
5 
4 
4 
5 
4 
5 

5 
4 

4 

4 
5 
4 
5 
4 
5 

4 
5 
4 

4 

5 
5 
4 
5 
4 
5 
4 
5 
4 
5 
5 
4 
5 
4 


97567 
563 
558 
554 
550 


60 

5 
58 
57 
56 




1 

2 
3 
4 
5 

6 
7 
8 
9 




1 




1 




1 




1 



1 






1 


















1 
1 


2 
3 


2 
3 

3 


2 
3 


2 

2 


2 

2 
3 


2 
2 
3 


2 
2 


447 
484 
520 
557 
593 


553 
516 
480 
443 
407 


902 
943 
984 
54025 
065 


098 
057 
016 
45975 
935 


455 
459 
464 
468 
472 


545 
541 
536 
532 
528 


55 

54 
53 
52 
51 


3 


3 


3 


3 

3 
4 
5 
5 


. 


1 
1 
1 


5 

5 

6 


5 
5 

C 


5 
5 

6 

6~ 


4 
5 
6 
6 


4 
5 

5 


4 

5 
5 


10 

11 
12 
13 
14 


629 
666 
702 
738 
774 


371 
334 

298 
262 
226 


106 
147 
187 
228 
269 


894 
853 

813 
772 
731 


477 
481 
485 
490 
494 


523 
519 
515 
510 
506 
" 501 
497 
492 
488 
484 


50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
t5 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 




10 

11 
12 
13 
14 


7 
8 
8 
9 
10 


7 





6 


6 


1 


8 
9 
9 


8 
8 
9 
10 
10 
11 
12 
12 


7 
8 
9 


7 

8 
8 
9' 
10 
10 
11 
11 


7 
8 
8 


7 

7 
8 




15 

16 
17 
18 
19 


811 
847 
883 
919 
955 


189 
153 
117 
081 
045 


309 
350 
390 
431 
471 


691 
650 
610 
569 
529 


499 
503 
508 
512 
516 


15 

16 
17 
18 
19 


10 

11 
12 
12 

13 


10 
11 
11 
12 
13 
13 
14 
15 
15 
16 


9 
10 

10 

11 
12 

12' 
13 
14 
14 
15 


9 

<* 
10 

10 

11 


8 
9 
10 
10 
11 


2 
2 


1 

1 
1 


29 

21 
22 
23 
24 


991 
52027 
063 
099 
135 


009 
47973 
937 
901 
865 


512 
552 
593 
633 
673 
~714 
754 
794 
835 
875 


488 
448 
407 
367 
327 


521 
525 
530 
534 
539 


47fl 
475 
470 
466 
461 


20 

21 
22 
23 
24 


14 

14 
15 
16 

1G 


13 
14 
14 
15 
10 
16 
17 
18 
18 
19 


12 
13 
13 
14 

14 


12 

12 
13 

13 
14 
"15" 
15 
16 
16 
17 


11 
12 

12 
13 

14 


2 
2 
2 
2 
2 


1 

1 
1 

2 
2 


25 

26 
27 
28 
29 


171 
207 
242 

278 
314 


829 
793 
758 
722 
686 


286 
246 
206 
165 
125 


543 
547 
552 
556 
561 


457 
453 
448 
444 
439 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
88 
39 


17 
18 
IS 
19 
20 
26" 
21 
22 
23 
23 
24" 
25 
25 
20 
27 

w 

28 
29 
29 
30 
31" 
31 
32 
33 
33 


17 
17 
18 
19 
19 
'20 
21 
21 
22 
23 


15 
16 
17 
17 

18 


15 
16 
16 
17 
17 
18" 
19 
19 
20 
20 
~2T 
22 
22 
23 
23 
24~ 
25 
25 
26 
26 
27~ 
28 
28 
29 
29 


14 
15 

15 
16 

16 


2 
2 
2 
2 
2 


2 
2 

2 
2 


30 

31 
32 
33 
34 


52350 
385 
421 
456 
492 


47650 
615 
579 
544 
508 


54915 
955 
995 
55035 
075 


45085 
045 
005 
44965 
925 


02565 
570 
574 
579 
583 
588 
592 
597 
601 
606 
610 
615 
619 
624 
628 


97435 
430 
426 
421 
417 


20 
20 
21 

21 
22 


18 
19 
20 
20 
21 
22" 
22 
23 
23 
24 
25 
25 
26 
27 
27 


18 

18 

19 
19 
20 


17 
18 
18 
19 
19 
20 
20 
21 
22 
22 
23 
23 
24 
24 
25 
26~ 
26 
27 
27 
28 


2 
3 
3 
3 
3 
~3~ 
3 
3 
3 
3 

~if 

3 
4 
4 
4 
~T 
4 
4 
4 
4 


2 

2 
2 
2 
2 


35 

36 
37 
38 
39 


527 
563 

598 
634 
669 


473 
437 
402 
366 
331 


115 
155 
195 
235 
275 


885 
845 
805 
765 
725 


412 
408 
403 
399 
394 


23 
24 
25 
25 

26 


23 

23 
24 
25 
25 
"26 
27 
27 
28 
29 


20 
21 
22 
22 
23 
23 
24 
24 
25 
26 

~6 

27 

27 

28 
29 


2 
2 
2 
3 
3 
T 
3 
3 
3 
3 
~3 
3 
3 
3 
3 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


705 
740 
775 
811 
846 


295 
260 
225 

189 
154 


315 
355 
395 
434 

474 


685 
645 
605 
566 
526 


390 
385 
381 
376 
372 


40 

41 
42 
43 

44 


27 

27 
28 
29 
29 
30~ 
31 
31 
32 
33 


881 
916 
951 
986 
53021 


119 
084 
049 
014 
46979 


514 
554 
593 
633 
673 


486 
446 
407 
367 
327 


633 
637 
642 
647 
651 


367 
363 
358 
353 
349 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


29 
30 
31 
31 
32 


28 
28 
29 
30 
30 


50 

51 
52 
53 

54 


056 
092 
126 
161 
196 


944 
908 
874 
839 
804 


712 
752 
791 
831 
870 


288 
248 
209 
169 
130 


656 
660 
665 
669 
674 


344 
340 
335 
331 
326 


10 

9 

8 
7 
6 


34 
35 
36 
36 
37 


33 

34 
35 

35 
30 


32 
33 
34 

34 
35 

w 

36 
37 
38 
38 


31 

31 
32 
33 
33 


30 
31 
31 
32 

32 


29 
30 
30 
31 
32 
32 
33 
33 
34 
34 


28 
29 

29 
30 
31 


4 
4 
4 
4 
4 


3 
3 
3 
4 
4 
4" 
4 
4 
4 
4 
"4 


55 

56 
57 
58 
59 


231 
266 
301 
336 
370 


769 
734 
699 
664 
630 


910 
949 
989 
56028 
067 


090 
051 
Oil 
43972 
933 


678 
683 
688 
692 
697 


322 
317 
312 
308 
303 


5. 
4 
3 
2 
1 


55 

56 
57 
58 
59 


38 
38 
39 
40 
40 


37 
37 
38 
39 

39 


34 
35 
35 
36 

36 


33 
34 
34 
35 

35 


31 
32 

32 
33 

33 


5 
5 
5 
5 
5 
~5 


60 


53405 


46595 


56107 


43893 


02701 


97299 





60 


41 
If 


40 

i<r 


39 

~wT 

Prc 


37 


36 

1<F 

iona 


35 
"35 

1 Pa 


34 
34 

rts 


' 


9. 

/cos 


d 
1' 


10. 

I sec 


9. 

f cot 


10. 

Ztan 


10. 

/ CSC 


d 

r 


9. 

I sin 




37 

port 


5 


4 



109 C 



70 

84 



20 C 



TABLE II 



159 C 



f 


I sin 
9. 


d 

1' 


I CSC 

10. 


Ztan 
9. 


d 


t cot 
10. 


Jsec 
10. 


d 


I COS 

9. 


f 


// 


40 


39 


Pro 
38 


"T 


onal 
35 


Pai 
34 


ts 
33 


5 


4 




1 

2 
3 

4 


53405 
440 
475 
509 
544 
578 
613 
647 
682 
716 


35 

35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
33 
34 

S 

34 
33 
34 
33 
34 
33 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 


46595 
560 
525 
491 
456 


56107 
146 
185 
224 
264 


39 
39 
30 
40 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
*9 
i9 
38 
39 
39 
39 
38 
39 
ffl 
38 
39 

39 
38 
39 
38 
39 
38 
38 
39 
38 
38 
30 
38 
38 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
37 
38 
38 
38 


43893 
854 
815 
776 
736 


02701 
706 
711 
715 
720 


5 
5 
4 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
5 
5 
4 
.") 
5 
5 
4 
5 
5 
4 

5 
5 
4 
5 
5 
5 
4 
5 
5 
5 
5 
4 
5 
5 
5 
5 
4 
5 
5 
5 
5 
4 

5 


97299 
294 
289 
285 
280 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 




1 

2 
3 
4 
5 

6 

7 
8 
9 




1 




1 



1 




1 



1 




1 



1 










~T 

1 
1 

2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 











1 
1 


2 
3 

3 
4 
5 
5 

6 


2 
3 
3 
4 
5 
5 
6 


2 
3 


2 

2 
3 
4 


2 

2 
3 
4 


2 
2 
3 

3 


2 
2 
3 

3 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 


422 
387 
353 
318 

284 


303 
342 

381 
420 
459 


697 
658 
619 
580 
541 


724 
729 
734 
738 
743 


276 
271 
266 
262 
257 


3 
4 


5 
6 


5 

6 


5 
5 
6 


5 
5 


4 
5 


751 
785 
819 
854 
888 


249 
215 
181 
146 
112 
"078 
043 
009 
45975 
941 


498 
537 
576 
615 
654 


502 
463 
424 
385 
346 
" 307 
268 
229 
190 
151 


748 
752 
757 
762 
766 


252 
248 
243 
238 
234 


50 

49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
*7 
36 


10 

11 
12 
13 
14 


7 


6 


6 


6 


6 


6 


1 
1 
1 
1 

1 
1 

1 

1 
1 
1 
1 
1 
1 

2 
2 
2 
2 
2 
2 
2 


8 
9 
9 
10 
11 
11 
12 
13 


8 
8 
9 
10 

10 

11 

12 

12 
13 
14 
14 
15 
16 
16 
17 
18 
18 
19 


S 
8 
9 


7 
8 
9 


7 
8 
S 


7 
7 
S 


7 
7 
8 
8 
9 
9 
10 
10 

11 

12 
12 

13 
13 
14 

14 
15 

15 
16 

16" 
17 
18 
18 
19 


15 

16 
17 
18 
19 
20 
21 
22 
23 
24 


922 
957 
991 
54025 
059 
093 
127 
161 
195 
229 


693 
732 

771 
810 
849 


771 
776 
780 
785 
790 
794 
799 
804 
808 
813 
" "818 
822 
827 
832 
837 


229 
224 
220 
215 
210 


15 

16 
17 
18 
19 


10 
10 
11 
11 
12 
13 
13 
14 
15 
15 


9 
10 

10 
11 
12 


9 

9 
10 

10 

11 


8 
9 
10 
10 

11 

ii" 

12 

12 
13 
14 
14 
15 
15 
16 
IB 


907 
873 
839 
805 
771 


887 
926 
965 
57004 
042 


113 
074 
035 

42996 
958 

~~9ftt 

880 
842 
803 
765 


206 
201 
196 
192 
187 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


13 

14 
15 

15 
16 
17" 
17 
18 
19 
19 


12 
13 
14 
14 
15 


12 
12 
13 

13 
14 
15" 
15 
16 
16 
17 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


263 
297 
331 
365 
399 


737 
703 
669 
635 
601 


081 
120 
158 
197 
235 


182 
178 
173 
168 
163 


35 

34 
33 
32 
31 


16 

16 
17 
18 
18 


15 
16 
17 
17 

18 


54433 
466 
500 
534 
' 567 


45567 
534 
500 
466 
433 


57274 
312 
351 
389 

428 


42726 
688 
649 
611 
572 


02841 
846 
851 
855 
860 
~ 865 
870 
874 
879 
884 


97159 
154 
149 
145 
140 


30 

29 
28 
27 
26 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 


20 
21 
21 
22 
23 


20 
20 
21 
21 
22 
23 
23 
24 
25 
25 


19 
20 
20 

21 
22 


18 
19 
20 

20 
21 


18 
18 
19 
19 
20 


17 
18 
18 
19 
19 


2 
3 
3 
3 
3 
3" 
3 
3 
3 
3 
~3~ 
3 
4 
4 
4 


2 

2 
2 
2 
2 


35 

36 
37 
38 
39 


601 
635 
668 
702 
735 


39 
365 
332 
298 
265 
231 
198 
164 
131 
097 


466 
504 
543 
581 
619 


534 
496 
457 
419 
381 


135 
130 
126 
121 
116 


25 

2/i 

21 
22 
21 
20 

19 
18 
17 
16 


23 
24 
25 
25 

26 

IT" 

27 
28 
29 
29 


22 
23 

23 
24 
25 

25 
26 
27 
27 

28 


22 
22 
23 

23 
24 


20 
21 
22 
22 
23 
23 
24 
24 
25 
26 


20 
20 
21 
22 
22 
23 
23 
24 
24 
25 


19 
20 

20 
21 

21 
~22 
23 
23 
24 
24 


2 
2 
2 
3 
3 
3 
3 
3 
3 
3 


40 

41 
42 
43 

44 


769 
802 
836 
869 
903 


658 
696 
734 
772 
810 


342 
304 
266 
228 
190 


889 
893 
898 
903 
908 
913 
917 
922 
927 
932 


111 
107 
102 
097 
092 


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25 
26 
20 
27" 
28 
28 
29 
29 
30 
31 
31 
32 
32 


18 
18 
19 
19 
20 
20 
21 
22 
22 
23 
23' 
24 
24 
25 
20 
26 
27 
27 
28 
29 
29" 
30 
30 
31 
32 


10 
17 
17 
18 
18 
19' 
19 
20 
20 
21 

21 
22 
22 
23 
23 
24 
25 
25 
26 
26 
27 
27 
28 
28 
29 
29 
30 
30 
31 
31 


10 
16 
17 
17 
18 
18 
19 
19 
20 
20 
~21 
21 
22 
22 
23 
23 
24 
24 
25 
25 
"26 
26 
27 
27 
28 


15 
16 
16 
16 
17 
18' 
18 
IS 
19 
20 
20 
20 
21 
22 
22 
22~ 
23 
24 
24 
24 


3 

3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
~4~ 
5 
5 
5 
5 
^5" 
5 
5 
5 
5 


588 r 
(>18' r 
648 f 
678: 
709! 

730: 

769: 
799: 
829: 
859' r 
"889^ 
919; 
949; 
979; 
59009; 


n 412 

382 
352 
" 322 

n 291 

"261 
231 
n 201 

n 171 

o 14 


079 
114 
150 
185 
221 
"256 
292 
327 
362 
398 


JO U1 

m 08 

o 5 

m 21 

"40991 


433 
468 
504 
539 
574 


45( 
45 
44, 
441 
435 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


31 

31 
32 
33 
33 


25 
26 
26 

26 
27 


24 

25 
25 
26 
26 


55 

5f 
57 
58 
59 


039; 
069; 
098; 

128; 
158: 


96 

a 93 

m 902 
" 872 

o 842 


609 
645 
680 
715 
750 


39 
355 
320 
285 
25(1 


571 
576 
581 
587 
592 


42' 
424 
41<3 
413 
408 




34 
35 
35 
36 

36 


33 
34 
34 
35 

35 


32 

33 
33 
34 

34 


28 
29 
29 
30 

30 


28 
28 
28 
29 
30 


27 
27 

28 
28 
29 


6 
6 

6 
6 


5 
5 
5 
5 
5 


60 


591 88 l 


40812 


62785 


37215 


03597 


96403 




60 


37 


36 


35 


32 
^2 

>por1 


31 


30 


29 


6 


5 


' 


9. 

JC08 


d 10. 

l I see 


9. 

Zcot 


d 
1 


10. 

Ztan 


10. 

1 esc 


d 

1 


9. 

1 sin 


' 





37 


36 


35 

Prc 


31 

iona 


30 

IPa 


29 

rts 


6 


5 


112 




67 

U7 





23 C 



TABLE II 



156 



' 


Ism 
9. 


d 


/CSC 

10. 


Han 
9. 


d 

1' 


/cot 
10. 


/sec 
10. 


d 

1' 

6 
5 
5 
6 
5 
6 
5 
5 
G 
5 
G 
5 
5 
6 
5 
6 
5 
6 
5 
6 
5 
5 
G 
5 
6 
5 
6 
5 
6 
5 
G 
5 
G 
5 
6 
5 
6 
5 
6 
5 
G 
5 
6 
6 
5 
6 
5 
6 
5 
fi 


1 COS 

9. 


/ 




n 


36 


35 


Prop 
34 


ortio 
30 




1 

2 
2 


nal Jr 
29 


'arts 
28 


6 


5 




1 

r 
t 

3 

4 


59188 
218 
247 
277 
307 


30 
29 
30 
30 
29 
30 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
29 
30 
29 
29 
30 
29 
29 
29 
29 
30 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
29 
28 
29 
<J 


40812 
782 
753 
723 
693 


62785 
820 
855 
890 
926 


35 
35 
35 
36 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
35 
34 
34 
34 
34 
35 
34 
34 
34 
34 
34 
34 
34 
34 
34 


37215 

180 
145 
110 
074 


03597 
603 
608 
613 
619 


96403 
397 
392 
387 
381 


60 

59 
58 
57 
56 




1 

2 
3 
4 



1 
1 
2 

2 



1 
1 
2 

2 



1 
1 
2 
2 





1 

1 
2 




1 
1 
2 
















5 

6 
7 
8 
9 
10 
11 
12 
13 
14 


336 
366 
396 
425 
455 


664 
634 
604 
575 
545 


961 
996 
63031 
066 
101 


039 
004 
36969 
934 
899 


624 
630 
635 
640 
646 


376 
370 
365 
360 
354 


55 

54 
53 
52 

51 


5 

6 


3 

4 


3 
4 


3 

3 


2 
3 


2 
3 
3 
4 

4 


2 
3 

3 
4 
4 




1 
1 
1 
1 
1 
1 
1 
1 
1 





1 
1 
f 
1 
1 

2 
2 
2 
2 
2 
2 
2 

2 
2 
2 
2 
2" 
3 
3 
3 
3 
3 
3 
3 
3 
3 


8 
9 
10 

11 
12 
13 
14 


5 

5 


5 
5 


5 
5 


4 

4 


484 
514 
543 
573 
602 


516 
486 
457 
427 
398 


135 
170 
205 
240 
275 


865 
830 
795 
760 
725 


651 
657 
662 
667 
673 


349 
343 
338 
333 
327 


50 

49 
48 
47 
46 



7 
7 

8 

8 


6 

6 

7 
8 
8 


6 
6 
7 

7 
8 


5 
6 
6 

6 

7 


5 

5 
6 

6 

7 


5 
5 

6 
6 

7 


15 

16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 


632 
661 
690 
720 
749 


368 
339 
310 
280 
251 


310 
345 
379 

414 
449 


690 
655 
621 
586 
551 


678 
684 
689 
695 
700 


322 
316 
311 
305 
300 


45 

44 
43 
42 
41 


15 

16 
17 
18 
19 


9 
30 
10 
11 

11 


9 

9 
10 

10 

11 

12 
12 
13 

13 
14 


8 
9 
10 
10 
11 


8 
8 
8 
9 
10 
10 
10 

11 

12 
12 


7 
8 
8 
9 
9 
10 
10 
11 

11 

12 


7 

7 
8 
8 
9 
9 
10 
10 

11 
11 

~12~ 
12 

13 
13 

14 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


778 
808 
837 
866 
895 


222 
192 
163 
134 
105 


484 
519 
553 
588 
623 


516 
481 
447 
412 
377 


706 
711 
716 
722 
727 


294 
289 
284 
278 
273 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 


12 
13 
13 
14 

14 
15 
lf> 
16 
17 
17 


11 
12 

12 
13 
14 


924 
954 
983 
60012 
041 


076 
046 
017 
39988 
959 


657 
692 
7^6 
761 
796 


343 
308 
274 
239 
204 


733 

738 
744 
749 
755 


267 
262 
256 
251 
245 


35 

34 
33 
32 
31 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


15 
15 
16 

16 
17 


14 
15 

15 
16 

16 


12 
13 
14 
14 
14 


12 

13 
13 
14 
14 


2 
3 
\\ 
3 
3 


30 

31 
32 
33 
34 


60070 
099 
128 
157 
186 


39930 
901 

872 
843 
814 


63830 
865 
899 
934 
968 


36170 
135 
101 
066 
032 


03760 
76G 
771 

777 
782 


96240 
234 
229 
223 

218 


30 

29 
28 
27 
26 


18 
19 
19 
20 

20 


18 
18 
19 
19 
20 

20 
21 
22 
22 
23 


17 
18 
18 
19 
19 


15 
16 
16 

16 
17 


14 
15 
15 
16 

16 


14 

11 
15 
15 
16 


3 

3 
3 
3 
3 


35 

36 
37 
38 
39 


215 
244 
273 
302 
331 


785 
756 
727 
698 
669 


64003 
037 
072 
106 
140 


35997 
963 
928 
894 
860 


788 
793 
799 
804 
810 


212 
207 
201 
196 
190 


25 

24 
23 
22 

21 


35 

36 
37 
38 

39 


21 
22 
22 
23 

23 


20 

20 
21 
22 
22 


18 
18 

18 
19 
20 


17 

17 
18 
18 
19 


16 
17 

17 
18 
18 


4 

4 
4 
4 
4 


40 

41 
42 
43 
44 


359 

388 
417 
446 
474 


641 
612 
583 
554 
526 


175 
209 
243 
278 
312 


825 
791 
757 
722 
688 


815 
821 
826 
832 
838 


185 
179 
174 
168 
162 


20 

19 
18 
17 
16 


40 

41 
42 
43 
44 


24 
25 
25 
26 

26 
27 
28 
28 
29 
29 


23 
24 

24 
25 
26 


23 
23 
24 

24 
25 


20 

20 
21 
22 
22 


19 
20 
20 
21 

21 


19 
19 

20 
20 
21 


4 

4 
4 
4 

4 


3 
3 
4 
4 
4 
4~ 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 
5 
5 
5 
5 


45 

46 
47 
48 
49 


503 
532 
561 
589 
618 


497 
468 
439 
411 
382 


346 
381 
415 
449 
483 


654 
619 
585 
551 

517 


843 
849 
854 
860 
865 


157 
151 
146 
140 
135 


15 

14 
13 
12 
11 
10 
9 
8 
7 
6 


45 

46 
47 
48 
49 


26 
27 

27 
28 
29 
29 
30 
30 
31 
32 


26 
26 
27 
27 

J*. 

28 
29 

29 
30 
31 


22 
23 
24 
24 
24 


22 
22 
23 
23 

24 


21 

21 
22 
22 
23 


4 
5 
5 
5 
5 


50 

51 
52 
53 

54 


646 
675 
704 
732 
761 


29 
29 
28 
29 
28 
29 
28 
29 
28 
28 


354 
325 
296 
268 
239 


517 
552 
586 
620 
654 


483 
448 
414 
380 
346 


871 
877 
882 
888 
893 


6 
5 
6 
5 
6 
6 
5 
6 
5 
6 


129 
123 
118 
112 
107 


50 

51 
52 
53 
54 


30 
31 
31 
32 

32 


25 
26 
26 

26 
27 


24 

25 
25 
26 
26 


23 

24 

24 
25 
25 
26 
26 
27 
27 
28 


5 

5 
5 
5 
5 


55 

56 
57 
58 
59 


789 
818 
846 
875 
903 


211 
182 
154 
125 
097 


688 
722 
756 
790 
824 


312 

278 
244 
210 
176 


899 
905 
910 
916 
921 


101 
095 
090 
084 
079 


5 

4 
3 
2 
1 


55 

56 
57 
58 
59 
60 


33 
34 
34 
35 

35 


32 

33 
33 
34 

34 


31 
32 

32 
33 

33 


28 
28 
28 
29 
30 


27 
27 

28 
28 
29 


6 
6 
6 
6 
6 
6 


80 


60931 


39069 


64858 


35142 


03927 


96073 





36 


35 


34 
~U 
Prop 


30 

To' 

ortio 


29 


28 
T8~ 
arts 


5 
"5 


* 


9. 

1 COS 


d 
1' 


10. 

{sec 


9. 

/cot 


d 
1' 


10. 

/tan 


10. 

Zcsc 


d 
1' 


9. 

/sin 


/ 


n 


36 


35 


29 

aalP 


6 


113 66 

68 



24 C 



TABLE II 



155 C 



9 



1 

2 
3 

4 


I sin d 
9. i 


icsc 
' 10. 


/ tan 
9. 


d 


I cot 
10. 


I sec d 
10. i 


I COS 

' 9. 


' 





34 


ti 
33 


oporl 
29 


ional 

28 


tart 
27 


s 
6 


5 


60931 
960 ? 
988^ 
61016, 
045 1 


,39069 
040 
! 012 
38984 
956 


64858 
892 
926 
960 
994 


34 
34 
34 
34 
34 
34 
34 
34 
34 
33 
34 
34 
34 
34 
33 
34 
34 
33 
34 
?'l 


35142 
108 
074 
040 
006 


03927 
933 I 
938 r; 
944 
950 ; 


96073 
,067 
062 
056 
050 


60 

59 
58 
57 
56 




i 

2 
3 

4 


' 
1 
1 
2 
2 



1 
1 
2 
2 




1 
1 
2 




1 
1 
2 




1 
1 
2 
















5 

6 

7 
8 
9 


073 

101 1 

129^ 
158 1 
186 1 


927 
! 899 

\ 871 
I 842 
s 814 


65028 
062 
096 
130 
164 


34972 
938 
904 
870 
836 


955 . 
961 ; 
966 \ 
972 
978 ( 


045 
039 
034 
' 028 

1 022 


55 

54 
53 
52 
51 


5 

6 

7 
8 
9 


3 

3 
4 
R 
5 


3 
4 

4 
5 


2 
3 

3 
4 

4 


2 
3 

3 
4 
4 


2 
3 
3 
4 
4 




1 





1 

1 
1 


10 

LI 

12 
13 
14 
15 

If 
L7 
18 
19 


2141 
242; 
270 1 
298; 
326 ' 
354. 
382; 
411* 
438 1 
466 g 
494, 
522 2 
550 * 
578 1 
606* 


8 786 
I 758 

I 73 
I 702 

8 G74 

"646 
o 618 
! 589 
1 562 
634 


197 
231 
265 
299 
333 


803 
769 
735 
701 
667 


983 . 
989 
995 ( 
04000 ; 
006 ( 


; 017 

! on 
: 005 
; ooo 

! 95994 


50 

49 
48 
47 
46 
15 
44 
43 
42 
41 


10 

11 
12 
13 
14 


c 

6 
7 

7 
8 


6 
6 
7 
7 

8 


5 

5 
6 
6 
7 


5 
5 

6 
6 
7 
"7" 

7 
8 
8 
9 


4 
5 
5 
6 

6 


1 

1 


1 
1 
1 
1 
1 


366 
400 
434 
467 
501 


634 
600 
566 
533 
499 


012 
018 ( 
023 
029 
035 


, 988 
] 982 
! 977 
? 971 
i 965 


15 

16 
17 
18 
19 


8 
9 
10 
10 
11 


8 
9 

9 
10 

10 


7 
8 
8 
9 
9 


7 
7 
8 
8 
9 


2 
2 
2 
2 
2 


1 

1 
1 

2 
2 


20 

21 
22 
23 
24 


8 5 6 

I 478 

ft 450 

8 422 
8 394 


535 
668 
602 
636 
669 


33 
34 
34 
33 
"44 


465 
432 
398 
364 
331 


040 
046 
052 
058 
063 


1 960 

954 
J 948 
J 942 

1 937 


40 

39 
38 
37 
36 
35 
34 
33 
32 
31 


20 

21 
22 
23 
24 


11 
12 

12 
13 
14 


11 

12 
12 
13 
13 


10 
10 

11 
11 

12 


9 
10 

10 

11 
11 


9 

9 
10 

10 

11 


2 

2 
2 
2 
2 


2 
2 
2 
2 
2 


25 

26 
27 
28 
29 


634, 
662* 
689 j 
717J 
745^ 


366 
! 338 

I 311 

283 

s 255 


703 
736 
770 
803 
837 


33 
34 
33 
34 
33 
34 
33 
34 
33 
34 
33 
SI 
34 
33 
33 
34 
33 
33 
33 
34 
33 
33 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
33 
33 

d 


297 
264 
230 
197 
163 


069 
075 
080 
086 
092 


, 931 
925 
1 920 
914 
J 908 


25 

26 
27 

28 
29 


14 
15 

15 
16 

10 


14 

14 
15 
15 
16 


12 

13 
13 
14 
14 


12 
12 
13 
13 

14 


11 

12 
12 
13 
13 
14 
14 
14 
15 
15 
"T6~ 
16 
17 
17 
18 
18 
18 
19 
19 
20 


2 
3 
3 
3 
3 


2 
2 
2 
2 
2 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


61773., 
800^ 
828! 
85( i 
883] 
91 1. 
939 
966, 
994 
62021 ; 


_ 38227 

8 2 
8 172 

! 144 

8^ J 1 7 
089 
! 061 

8 34 
, 006 

I 37979 
'""951 

8 924 
I? 896 

8(59 
* 84 


65870 
904 
937 
971 
66004 
038 
071 
104 
138 
171 


34130 
096 
063 
029 
33996 
962 
929 
89( 
862 
829 
796 
762 
729 
696 
663 


04098 
103 
109 
115 
121 
127 
132 
138 
144 
150 


, 95902 
; 897 
J 891 
885 
b 879 
' 873 
? 868 
862 
85f 
5 850 


30 

29 
28 
27 
26 
25 
24 
23 
22 
21 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


17 
IS 
18 
19 
19 
20 
20 
21 
22 
22 


16 
17 
18 
18 
19 
^9~ 
20 
20 
21 

21 
22 
23 
23 
24 
24 


14 
15 
15 
16 

16 

TT 

17 
18 
18 
19 
19 
20 
20 
21 
21 


14 

14 
15 

15 
16 
~ 16 " 
17 
17 
18 
18 
19 
19 
20 
20 
21 


3 

3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 


2 
3 
3 
3 
3 
3 
3 
3 
3 
3 


40 

41 
42 
43 
44 


04< * 
076; 
104' 

131 ; 

159? 


204 
238 
271 
304 
337 


156 
161 
167 
173 
179 


' 844 
J 839 
~ 833 
" 827 

6 821 


20 

19 

18 


40 

41 
42 
43 
44 


23 
24 

24 
25 


3 
3 
4 
4 
4 


45 

4( 
47 
48 
49 


186 
214' 
241? 
268' 
296? 


814 
J 786 
J 759 

8 732 

* 704 


371 
404 
437 
470 
503 


629 
596 
563 
530 
49 


185 
190 
196 
202 
208 


, 815 
? 810 
1 804 
? 798 
! 792 


i 


45 

46 
47 

48 
49 


26 
26 
27 
27 

28 


25 

25 
26 

26 
27 


22 
22 
23 
23 

24 


21 

21 
22 
22 
23 


20 
21 
21 
22 
22 


4 
5 
5 
5 
5 


4 
4 
4 
4 
4 


50 
51 
52 
53 
54 


323; 

350 ? 
377 - 
405' 
432 f 


1 677 
^ 650 
I 623 
! 59S 
J 568 


537 
570 
603 
636 
669 


463 
430 
397 
364 
33 


214 
220 
225 
231 
237 


6 78G 
J 780 

A 77S 
! 769 

J 763 




50 

51 
52 
53 
54 


28 
29 
29 
30 
31 


28 
28 
29 
29 
30 


24 

25 
25 
26 
26 


23 
24 
24 
25 
25 


22 
23 
23 
24 

24 


5 

5 
5 
5 
5 


4 
4 
4 
4 
4 


55 

56 
57 

58 
59 


459; 
486? 
513? 
541? 
568? 


7 54 

J 514 

8 48? 

S 459 
7 432 


702 
735 
768 
801 
834 


298 
265 
232 
199 
166 


243 
249 
255 
261 
267 


6 75? 
I 751 

A 745 
A 739 

? 733 




55 

56 
57 

58 
59 


31 
32 

32 
33 

33 


30 
31 

31 
32 

32 


27 
27 

28 
28 
29 


26 
26 
27 
27 

28 


25 
25 

26 
26 

27 


6 
6 
6 
6 
6 


5 
5 
5 
5 
5 


SO 


62595 ' 


37405 


66867 


33133 


04272 


95728 


( 


60 


34 


33 


29 


28 


27 


6 


5 


' 


1 COS 


d 10. 

l' Z sec 


9. 

Zcot 


10. 

Ztan 


10. 

I CSC 


d 9. 

i' 1 sin 


' 




34 


33 

F 


29 

fopoi 


28 
rtioiu 


27 

ilPai 


6 

ts 


5 


114 


65 

69 





25 C 



TABLE II 



154 C 



' 


I sin 
9. 


d 

1' 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 

i 

27 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 


I CSC 

10. 


Jtan 
9. 
66867 
900 
933 
966 
999 


d 

i' 

33 
33 
33 
33 
33 
33 
33 
33 
32 
33 
33 
33 
33 
32 
33 
33 
33 
32 
33 
33 
32 
33 
33 
32 
33 
32 
33 
33 
32 
33 
32 
33 
32 
33 
32 
32 
33 
32 
33 
32 
3,' 
33 
32 
32 
33 
32 
32 
32 
33 
32 
32 
32 
32 
33 
32 
32 
32 
32 
32 
32 


I cot 
10, 


1 sec 
10. 


d 

r 

6 
6 
6 
6 

6 
6 
6 
6 

5 
6 

I) 
(} 

G 
(i 
G 
fj 


1 COS 

9. 


60 

59 
58 
57 
56 






1 

2 
3 
t 
5 


7 

8 
9 


33 

~~ 

1 

1 

2 


P 
32 

(f 
1 
1 
2 
2 


ropor 
27 


tiona 
26 


IPar 
7 








ts 
6 








1 
1 

1 
1 


5 

~~6~ 




o 





i 
i 
i 




1 

2 
3 
4 


62695 
622 
649 
676 
703 


37405 
378 
351 
324 
297 


33133 
100 
067 
034 
001 


04272 
278 
284 
290 
296 


95728 
722 
716 
710 
704 





1 

1 

2 

3 
3 

4 
4 



() 

1 

2 
2 
3 
3 
3 
4 


5 

6 
7 
8 
9 


730 
757 

784 
811 
838 


270 
243 
216 
189 
162 


67032 
065 
098 
131 
163 


32968 
935 
902 
869 
837 


302 
308 
314 
320 
326 


098 
602 
686 
680 
674 


55 

54 
53 
52 
51 
50 
49 
48 
47 
40 


3 

3 
4 

4 
5 


3 
3 
4 
4 
5 


1 
1 
1 

1 
1 
1 
1 
1 

2 
2 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


865 
892 
918 
945 
972 


135 
108 
082 
055 
028 
"Obi 
36974 
948 
921 
894 


196 
229 
262 
295 
327 


804 
771 
738 
705 
673 
640 
607 
574 
542 
509 


332 
337 
343 
349 
355 


6G8 
003 
657 
651 
645 


10 

11 
12 
13 
14 
15 
10 
17 
IS 
19 


6 
6 

7 
7 

s 


5 
6 

G 

7 

7 


4 
5 

5 
6 

n 


4 
5 
5 
6 
6 


1 
1 
1 
1 
1 


i 
i 
1 

i 
i 


999 
63026 
052 
079 
106 


360 
393 
426 
458 
491 


361 
367 
373 
379 
385 


039 
633 
627 
621 
615 


45 

14 
43 
42 
41 


8 
9 

H 
10 

10 


S 
I) 
9 
10 
10 


7 
7 

8 
8 
<) 


6 
7 

7 
8 
S 


2 
2 
2 

2 




2 

2 
2 
2 


i 
i 

2 


90 
21 

22 
23 
24 
25 

26 
27 
28 
29 


133 
159 
186 
213 
239 
266 
292 
319 
345 
372 


867 
841 
814 
787 
761 
734 
708 
681 
655 
628 


524 
556 

589 
622 
654 
687 
719 
752 
785 
817 


476 
444 
411 

378 
346 
313 
281 
24S 
215 
183 


391 
397 
403 
409 
415 
421 
427 
433 
439 
445 


G 

6 
G 

G 
f> 
G 
G 

: 



s 

6 

G 

J 


6 
6 
ft 
G 
(> 


G 
fi 

' 

; 

G 
6 
6 

G 
7 
G 
G 
6 


009 
003 
597 
591 
585 
" 579 
573 
507 
501 
555 


40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
2:5 
22 
21 
20 
19 
18 
17 
1(5 
15 
14 
13 
12 
11 

To 

<j 

i 
( 




20 

21 
22 
23 
24 
"25 
20 
27 
2S 
29 


11 
12 
12 
13 
13 
14 
11 
15 
15 
16 


11 
11 

12 

12 
13 
13 
14 

11 
15 

lf> 


9 

<J 
10 

10 

11 

11 

12 
12 
13 
13 


i) 
9 
10 
10 

10 

TT 

11 

12 

13 


2 
2 
3 
3 
3 
~~3 
3. 
3' 
3 
:i 


2 

2 
2 
2 
2 

S 

3 
3 
3 


2 

2 

2 
2 

2 
2 

3 
3 
3 
3 


30 

31 
32 
33 
34 


63398 
425 
451 
478 
504 


36602 
575 
549 
522 
496 


67850 
882 
915 
947 
980 


32150 
118 
085 
053 
020 


04451 

457 
403 
409 
475 
481 
487 
493 
500 
500 


95549 
543 
537 
531 
525 


30 

31 
32 
33 
34 


10 
17 
IS 
18 
19 


1C. 
17 
17 

is 
18 


14 
14 
14 
15 
15 
10 
16 
17 
17 
is 
18 
is 
19 
19 

20 

20 
21 

21 

22 
22 
22 
23 
23 
24 
24 
25 
25 
20 
26 
27 
27 


13 

[I 

14 
15 
15 
1(> 
16 
10 
17 


4 
4 
4 
4 
4 


3 

3 
3 
3 


35 

36 
37 
38 
39 


531 
557 
583 
610 
636 


469 
443 
417 
390 
364 


68012 
044 
077 
109 
142 


3198S 
95(5 
923 
891 
858 
82(5 
794 
761 
72!) 
697 


519 
513 

507 
500 
494 
488 
482 
470 
470 
404 


,35 

30 
37 
38 
39 


19 
20 

20 

21 

21 


19 
19 
20 
20 
21 


4 
4 
4 
4 
5 
5 
5 
5 
5 
5 


4 

4 
4 
4 
J 
4 
4 
4 
4 
4 


3 
3 

3 
3 
3 
3 

3 
4 
4 
4 

"l 
4 
4 
4 
4 
4 
4 
4 
4 
4 


40 

41 
42 
43 
44 


662 
689 
715 
741 
767 


338 
311 
285 
259 
233 


174 
206 
239 
271 
303 


512 
518 
524 
530 
536 


40 

41 
42 
43 
44 


22 
23 
23 
24 
24 


21 
22 

'>'> 

2,3 

23 


is 
18 
19 
19 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


794 
820 
846 
872 
898 


206 
180 
154 
128 
102 


336 
368 
400 
432 
465 


664 
632 
600 
568 
535 
503 
471 
439 
407 
374 


542 

548 
554 
560 
566 


458 
452 
440 
440 
434 


45 

40 
47 
48 
49 


25 

25 
26 
20 
27 
28" 
28 
29 
29 
30 
,30 
31 
31 
32 
32 


2-1 
25 
25 
20 
28 
27 
27 
28 
2S 
29 
"i?9 
30 
30 
31 
31 
32 


20 
20 
20 
21 

21 


5 
5 
5 





4 
5 
5 
5 
5 


924 
950 
976 
64002 
028 


076 
050 
024 
35998 
972 


497 
529 
561 
593 
626 


573 

579 
585 
591 
597 


427 
421 
415 
409 
403 
397 
391 
3S4 
378 
372 


50 

51 
52 

r>3 

f>4 
55 

56 
57 

58 
59 


22 

22 

23 
23 

23 


(i 
6 

(i 
(i 

(i 


5 

5 
5 
5 
5 


55 

56 
57 
58 
59 
60 

/ 


054 
080 
106 
132 
158 
64184 


946 
920 

894 
868 
842 


658 
690 
722 
754 
786 


342 
310 
278 
24( 
214 


603 
009 
616 
622 
628 


5 


24 

24 
25 
25 
26 
26 
26 
tiona 


(> 

7 
7 
7 
7 


(i 
(1 

6 
<> 


5 
5 
5 
fi 
5 


35816 


68818 


31182 


04634 


9536(J 





60 


33 


. 5 


9. 

1 COS 


d 
1' 


10. 

Zsec 


9. 

Zcot 


d 


10. 

Ztan 


10. 

1 CMC 


d 
1' 


9. 

/ sin 


' 




33 


32 

P 


27 

ropoi 


7 

IPar 


6 

ts 


5 


115 64 

70 





26 C 



TABLK II 



153 f 





1 

2 
3 

4 


I sin 
9. 


d 

1' 

26 
26 
26 
26 
25 
26 
26 
26 
26 
25 
26 
26 
25 
26 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
25 
25 
26 
25 
25 
25 
26 
25 
25 
25 
25 
25 
25 
25 
24 
25 
25 
25 
25 
25 


I CSC 

10. 

35816 
790 
764 
738 
712 


/tan 
9. 


d 

1' 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
31 
32 
32 
32 
J2 
31 
32 
32 
32 
31 
32 
32 
31 
32 
32 
31 
32 
31 
32 
32 
31 
32 
31 
32 
31 
32 
31 
32 
31 
32 
31 
31 
32 
31 
32 
31 
31 
32 
31 
31 
32 
31 
31 
31 
32 

d 

r 


Jcot 
10. 

3TT82 
150 
118 
086 
054 


/ sec 
10. 

04634 
640 
646 
652 
659 
665 
671 
677 
683 
690 
696 
702 
708 
714 
721 
727 
733 
73<) 
746 
752 


d 


/cos 
9. 
95366 
360 
354 
348 
341 


' 




tt 


32 


Pi 
31 


| 

2 

~rr 

3 

4 
4 
5 
5 


6 

7 
7 
8 
8 
9 
9 
10 


V 





1 

1 

2 

2' 
3 
3 

3 
4 
~4 
5 
5 
6 
6 
6 
7 

I 

8 
9 
9 
10 
10 
10 


tiona 
25 


Part 
24 


s 
7 


6 






~~Q~ 
1 

1 
1 
1 
1 
1 
1 
1 
1 

2 
2 
2 

2 
2 


64184 
210 
236 
262 
288 


68818 
850 
882 
914 
946 


6 
6 
G 
7 
f> 
6 

(i 
7 
C 


6 
7 

6 
6 

; 

<> 

6 

i 



7 
(i 
fi 

6 
6 

; 




60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 




i 

2 

a 

4 
5 

6 
7 
8 
9 

~io 

11 

12 
13 
14 
15 

16 
17 
18 
19 




2 

3 
3 

5 

6 

(i 

7 

8 
9 
9 
10 
10 
11 " 
11 

12 

12 
13 





1 

2 
"V 
2 
3 
3 
4 
4 
5 
5 
5 
6 

V 

7 

? 
8 
8 

8 
9 

9 
10 
10 





1 
1 

"2 








1 
1 
1 
1 

1 
1 
1 

1 

2 
2 

~Y 

2 
2 

2 
2 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


313 
339 
365 
391 
417 
442 
468 
494 
519 
545 
57i 
596 
622 
647 
673 


687 
661 
635 
609 
583 
558 
532 
506 
481 
455 
4~29 
404 
378 
353 
327 


978 
69010 
042 
074 
106 
"138 
170 
202 
234 
266 
" 298 
329 
361 
393 
425 
457 
488 
520 
552 
584 


022 
30990 
958 
926 
894 
862 
830 
798 
766 
734 
702 
671 
639 
607 
575 
543 
512 
480 
448 
416 


335 
329 
323 
317 
310 
304 
298 
292 
286 
271) 
273 
267 
201 
254 
248 
242 
236 
229 
223 
217 
211 
204 
198 
192 
1S5 
95179 
173 
167 
160 
154 


2 

3 
3 
4 
4" 

4 
5 

5 
6 

IT 

6 
7 
7 

8 


698 
724 
749 
775 
800 


302 
276 
251 
225 
200 


758 
764 
771 
777 
783 
789 
796 
802 
808 
815 
04821 
827 
833 
840 
846 


20 

21 
22 
23 
24 


10 

11 

11 
12 


8 
8 
9 

9 
10 


2 
2 
3 
3 
3 


2 

2 
2 
2 
2 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


826 
851 
877 
902 
927 
64953 
978 
65003 
029 
054 


174 
149 
123 
098 
073 
35047 
022 
34997 
971 
946 


615 
647 
679 
710 
742 
69774 
805 
837 
868 
900 


385 
353 
321 
290 
258 
30226 
195 
163 
132 
100 


35 

34 
33 
32 
31 
30 
29 
28 
27 
26 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


13 
14 

14 
15 

15 
16 
17 
17 

IS 
18 


13 

13 
14 

14 
15 
IB 
16 
17 
17 
IS 


11 

11 

12 
12 
13 
13 
13 
14 
14 
15 
15 
16 

16 

16 
17 


10 

11 

11 

12 
12 
12~ 
13 
13 
14 
14 


10 

10 

11 
11 

12 
12" 

12 
13 

13 
14 


3 
3 

3 
3 
3 
~~4~ 
4 
4 
4 
4 


2 
3 
3 
3 
3 
~~3 
3 
3 
3 
3 


35 

36 
37 
38 
39 


079 
104 
130 
155 
180 


921 
'896 
870 
845 
820 


932 
963 
995 
70026 
058 


068 
037 
005 
29974 
942 


852 
859 
865 
871 
878 


7 
(i 

; 

(i 
d 

7 
6 
7 
6 

7 

(j 
7 
6 
7 
C 
IS 
7 

5 

6 
G 

7 


148 
141 
135 
129 
122 


25 

24 
23 
22 

21 




35 

30 
37 
38 
39 


19 
19 
20 
20 
21 


18 

19 
19 

20 

20 

21~ 
21 
22 
22 
23 
23 
24 
24 
25 
25 


15 
15 
15 
16 

16 


14 

14 
15 
15 
16 
16~ 
16 
17 
17 
18 

~18~ 
18 
19 
19 
20 


4 
4 
4 
4 
5 


4 
4 
4 
4 
4 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


205 
230 
255 
281 
306 
331 
356 
381 
406 
431 


795 
770 
745 
719 
694 


089 
121 
152 
184 
215 


911 

87<J 
848 
816 
785 


884 
890 
897 
903 
910 


116 
110 
103 
097 
090 
084 
078 
071 
065 
059 
"052 
046 
039 
033 
027 


20 

19 
18 
17 
16 


40 

41 
42 
43 
44 
45 ~ 
40 
47 
48 
49 


21 
22 

22 
23 
23 
24 
25 
25 
20 
26 


17 
18 
18 
19 
19 


17 
17 
18 
18 
18 
~19~" 
19 
20 
20 
20 


5 
5 
5 
5 

5 

" 5 
5 
5 
6 
6 


4 

4 
4 
4 
4 


669 
644 
619 
594 
569 


247 
278 
309 
341 
372 
~404 
435 
466 
498 
529 
560 
592 
623 
654 
685 


753 
722 
691 
659 
628 


916 
922 
929 
935 
941 


15 

14 
13 
12 
11 


20 
20 
20 
21 

21 


4 
5 
5 
5 
5 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
00 


456 
481 
506 
531 
556 


544 
519 
494 
469 
444 
420 
395 
370 
345 
320 


596 
565 
534 
502 
471 
"440 
408 
377 
346 
315 
29283 


948 
954 
961 
967 
973 
~980 
986 
993 
999 
05005 
05012 


10 

9 

8 
7 
() 

A 

4 

2 

1 

6 


50 

51 
52 
53 
54 


27 
27 

28 
28 
29 
29 
30 
30 
31 
31 


26 

20 
27 

27 
28 


22 
22 
23 
23 

23 


21 

21 
22 
22 
22 
23 
23 
24 
24 
25 


20 

20 
21 
21 
22 
22 
22 
23 
23 
24 


6 
6 

6 
6 
6 


5 

5 
5 
5 
5 


580 
605 
630 
655 
680 
65705 


1)20 
014 
007 
001 
94995 
94988 


55 

56 
57 

58 
59 


28 
29 
29 
30 

30 


24 

24 
25 
25 
26 
26 
26~ 
ropoi 


6 
7 
7 
7 

7 


6 
6 
C 
6 
6 


34295 


70717 


60 


32 


31 


25 


24 


7 


6 


9. 

1 COS 


d 

1' 


10. 

/ sec 


9. 

Jcot 


10. 

Han 


10. 

/ CSC 


d 
1' 


9. 

/ sin 


f 






32 


31 

P 


25 

tiona 


24 

IPar 


7 

ts 


6 


116 C 63 

71 





TABLE II 



152 C 





1 

2 
3 

4 


/sin 
9. 


d 
1' 

24 
25 
25 
25 
24 
25 
25 
24 
25 
25 
24 
25 
24 
25 
25 
24 
25 
24 
25 
24 
24 
25 
24 
25 
24 
24 
25 
24 
24 
25 
24 
24 
24 
24 
25 
24 
24 
24 
24 
24 
24 
25 
24 
24 
24 
24 
24 
24 
24 
23 
24 
24 
24 
24 
24 
24 
24 
23 
24 
24 


ICBC 

10. 


I tan 
9. 


d 

31 
31 
31 
31 
32 
31 
31 
31 
31 
31 
31 
31 
31 
32 
31 
31 
31 
31 
31 
31 
31 
31 
30 
31 
31 
31 
31 
31 
31 
31 
31 
30 
31 
31 
31 
31 
30 
31 
31 
30 
31 
31 
31 
30 
31 
31 
30 
31 
30 
31 
31 
30 
31 
30 
31 
30 
31 
30 
31 
30 

d 

r 


I cot 
10. 


2 sec 
10. 


d 
1' 

6 
7 
6 

6 

7 
C 
7 
6 

6 
6 

6 
7 
6 
7 
7 
C 
7 
6 
7 
6 
7 
6 
7 
6 
7 
7 
6 
7 
6 
7 
G 
7 
7 
G 
7 
6 
7 
7 
G 
7 
7 
G 
7 
7 
6 
7 
7 
G 
7 
7 
6 
7 
7 
G 
7 
7 
7 


1 COS 

9. 


60 

59 
58 
57 
56 




n 

~lf 

i 
2 
3 
4 


39 


1 


31 


1 


SB* 





ortio 
95 


nail 3 
94 





arts 
93 



1 
1 

2 


7 


6 


65705 
729 

754 
779 
804 


34295 
271 
246 
221 

196 


70717 
748 
779 
810 
841 


99283 
252 
221 
190 
159 


05012 
018 
025 
031 
038 


94988 
982 
975 
969 
962 




















2 
9 


2 
9 


2 
9 


1 
2 


1 
2 
9 
2 
3 
3 


5 

6 
7 
8 
9 


828 
853 
878 
902 
927 


172 
147 
122 
098 
073 


873 
904 
935 
966 
997 


127 
096 
065 
034 
003 


044 
051 
057 
064 
070 


956 
949 
943 
936 
930 


55 

54 
53 
52 
51 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 

Ts 

16 
17 
18 
19 
90 
21 
22 
23 
24 


3 
3 
4 
4 
5 


3 
3 

4 
4 

5 


2 
3 
4 
4 


9 

2 
3 

3 


9 

2 
3 
3 


1 
1 
1 
1 




1 
1 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
90 
21 
22 
23 
24 


952 
976 
66001 
025 
050 
075 
099 
124 
148 
173 
" 197 
221 
246 
270 
295 


048 
024 
33999 
975 
950 


71028 
059 
090 
121 
153 


98972 
941 
910 
879 
847 


077 
083 
089 
096 
102 


923 
917 
911 
904 
898 


50 

49 
48 
47 
46 
45 
44 
43 
42 
41 


5 
6 
6 
7 

7 
8 
9 
9 

10 

11 
n 
11 
19 

12 
13 


5 

6 
6 
7 
7 
8 
8 
9 
9 

1 

10 
11 
11 
19 

12 


5 
6 
6 

C 
7 
8 
8 
8 
9 
10 

~w 

10 

n 

12 
19 


4 

5 
5 

5 
6 
6 
7 
7 
8 
8 
8 
9 
9 
10 
10 


4 

4 
5 
5 
6 
6 
6 
7 
7 
8 
8 
8 
9 
9 
10 


4 

4 
5 
5 
5 
6 
6 
7 
7 
7 

"s 

8 
8 
9 
9 


1 
1 
1 

2 
2 
~~2~ 
2 
9 

2 
2 
~2~ 
2 
3 
3 
3 


~T 

2 
2 
2 
2 
~9~ 
2 
2 
2 
2 


925 
901 
876 
852 

827 


184 
215 
246 
277 
308 


816 
785 
754 
723 
692 
661 
630 
599 
569 
538 


109 
115 
122 
129 
135 
142 
148 
155 
161 
168 


891 
885 
878 
871 
865 


803 
779 
754 
730 
705 


339 
370 
401 
431 
462 


858 
852 
845 
839 
832 
"826 
819 
813 
806 
799 


40 

39 
38 
37 
36 


95 

26 
27 

28 
29 


319 
343 
368 
392 

416 


681 
657 
632 
608 
584 


493 
524 
555 

586 
617 
71648 
679 
709 
740 
771 


507 
476 
445 
414 
383 


174 
181 
187 
194 
201 


35 

34 
33 
32 
31 
30 
29 
28 
27 
26 
95 
24 
23 
22 
21 


95 

26 
27 
28 
29 


13 
14 

14 
15 

15 
~W 
17 
17 

18 
18 


13 

13 
14 
14 
15 
16 
16 
17 
17 
18 
18 
19 
19 
20 
90 


12 
13 
14 
14 
14 


10 

11 

11 

12 
19 


10 

10 

11 
11 

12 


10 
10 

10 

11 
11 


3 
3 

3 
3 
3 
4 
4 
4 
4 
4 


2 
3 
3 
3 
3 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 


66441 

465 
489 
513 
537 


33559 
535 
511 
487 
463 


98352 
321 
291 
260 
229 
198 
167 
137 
106 
075 


05207 
214 
220 
227 
233 
240 
247 
253 
260 
266 


94793 

786 
780 
773 
767 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


15 
16 
16 
16 
17 
18 
18 
18 
19 
20 
90 
20 
21 
22 

*L 

22 
23 
24 
94 
24 
25 
26 
96 
26 
27 
~28~ 
98 
28 
29 
30 
"30" 


12 
13 
13 
14 
14 
15 
15 
15 
16 
16 


19 

12 
13 

13 

14 


12 
19 
12 
13 
13 


3 

3 
3 
3 
3 
4 
4 
4 
4 
4 


562 
586 
610 
634 
658 


438 
414 
390 
366 
342 


802 
833 
863 
894 
925 


760 
753 
747 
740 
734 


19 
19 
20 
90 
91 


14 

14 
15 
15 
16 


13 
14 
14 
15 
15 


4 
4 
4 
4 
5 


682 
706 
731 
755 
779 
803 
827 
851 
875 
899 
"922 
946 
970 
994 
67018 


318 
294 
269 
245 
221 


955 
986 
79017 
048 
078 
109 
140 
170 
201 
231 


045 
014 
97983 
952 
922 


273 
280 
286 
293 
300 


727 
720 
714 
707 
700 
694 
687 
680 
674 
667 
660 
654 
647 
640 
634 


90 

19 

18 
17 
16 


40 

4t 
42 
43 
44 


21 
99 
22 
93 

23 


21 
91 
22 
99 
23 
93 
94 
24 
95 
25 


17 
17 

18 
18 
18 
19 
19 
20 
90 
20 
91 
21 
22 
99 
22 


16 

16 
17 

17 
18 

~W 

18 
19 
19 

20 


15 
16 
16 
16 
17 
17 
18 
18 
18 
19 
19 
20 
90 
20 
21 
91 
21 
99 
22 
23 


5 
5 
5 
5 

5 


4 

4 
4 
4 
4 


197 
173 
149 
125 
101 


891 
860 
830 
799 
769 


306 
313 
320 
326 
333 


15 

14 
13 
12 
11 
10 
9 
8 

6 


45 

46 
47 
48 
49 


24 
25 
95 
26 
96 


5 
5 
5 
6 
6 
~T 
6 
6 
6 
6 


4 
5 
5 
5 
5 


078 
054 
030 
006 
32982 
958 
934 
910 
887 
863 


262 
293 
323 
354 
384 


738 
707 
677 
646 
616 


340 
346 
353 
360 
366 


50 

51 
52 
53 
54 


27 
97 
98 
28 
99 
29 
30 
30 
31 
31 
32 


96 

26 
97 

27 

98 


90 
20 
91 
21 
22 
99 
22 
23 
93 
24 


5 

5 
5 
5 
5 


55 

56 
57 
58 
59 


042 
066 
090 
113 
137 


415 
445 
476 
506 
537 
79567 


585 
555 
524 
494 
463 


373 
380 
386 
393 
400 


627 
620 
614 
607 
600 


5 

<j 

i. 

] 




55 

56 
57 
58 
59 
60 


28 
99 
29 
30 

30 
"31 


93 

23 
24 
94 
25 
95 


6 
7 

7 
7 
7 


6 
6 
6 
6 
6 


60 


67161 


39839 


97433 


05407 


94593 


94 


93 
-93 

Parts 


7 
~7 


6 


f 


9 

1 COS 


d 

1' 


10. 

Zsec 


9. 

I cot 


10. 

itan 


10. 

/ CSC 


d 
1' 


9. 

Jsin 


f 


n 


39 


31 


30 

Proj 


95 

Dortit 


94 

mal 


6 


117 62 

72 



28 C 



TABLE II 



151 C 



, Tsm 
9. 


/ CSC 

10. 


tan d 
9. i 


I cot 
10. 


soc d 
10. i' 


I COS / 

9. 


n 


31 


1 
30 


J ropo 
29 


rtion 
4 




1 

1 

2 


al 1^ 
x3 


arts 
M 



1 

1 

1 


7 


6 


0671G1 
1 185 
2 208 
3 232 
4 256 


32839 
815 
792 
1 768 

1 744 


2567 
598*1 
628 3 3 
659*! 
_68_9 
720 * 
750? 
780 f 

81 l v 
841 f 

872 _. 
902 J 
932* 

96 4 
993* 

-3023 
054 3 . 
084 '' 
114* 
144" 


27433 
402 
372 
341 
311 


5407 R 
413? 
420 ; 

427 o 
433? 


94593 60 
587 59 
58058 
57357 
567 56 




1 

2 
3 
4 



1 
1 
2 
2 





1 

2 
2 




1 
1 
2 





1 

1 

2 
















5 280 
6 303 
7 327 
8 350 
9 374 


, 720 
J 697 
673 
650 
626 


280 
250 
220 
189 
159 


440 \ 
447 ] 

454 o 
460 2 

467 1 


560 55 
55354 
54653 
54052 
53351 


5 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


3 
3 
4 
4 
5 
5 
6 
6 
7 
7 
8 
8 
9 
9 
10 


2 
3 
4 
4 

4 
~~& 

6 
6 


7 
8 
8 
8 
9 
10 


2 
3 
3 
4 

4 

~T 

5 
6 

6 
7 
7 
8 
8 
9 
9 


2 

2 
3 
3 
4 
4 
4 
5 
5 
6 
6 
6 
7 
7 
8 


2 

2 
3 
3 

3 

4 
5 
5 
5 
6 
6 
7 
7 
7 

T 

8 
8 
9 
9 


2 

2 
3 
3 
3 
4 
4 
4 
5 
5 
6 
6 
6 
7 
7 


1 
1 
1 

2 
2 
2 
2 
2 

2 
2 

T 
2 
3 
3 
3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
5 





10 398 

11 421 
12 445 
13 468 
14 492 
15 5f5 
16 539 
17 562 
18 586 
19 609 


602 
579 
555 
532 
508 
, 485 
* 461 
, 438 
J 414 
I 391 


128 
098 
068 
037 
007 
26977 
946 
916 
886 
856 


474 
481 
487 
494 
501 
508 
515 
521 
528 
535 
~542 
549 
555 
562 
569 

"57fi 
583 
590 
596 
603 


526 50 

51949 
51348 
50647 
49946 
492 45 
48544 
47943 
47242 
465 41 


~~2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


20 633 

21 656 
22 680 
23 703 
24 726 
25 750 
26 773 
27 796 
28 820 
29 843 


, 367 
, 344 
* 320 
, 297 
I 274 


176 1 
205 * 

235? 
265, 
295^ 
326* 
356^ 
386* 
416'* 
446, 


825 
795 
765 
735 
705 


45840 
451 39 
44538 
43837 
431 36 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


10 

11 

11 

12 

12 
13 
13 
14 

14 
15 


10 

10 

11 

12 
12 

12 
13 
14 
14 

14 


10 
10 

11 
11 

12 
12 
13 
13 
14 
14 


8 
8 
9 
9 
10 
10 
10 

11 
11 

12 
12 

12 
13 

13 
14 
14 
14 
15 
15 
16 


7 
8 
8 
8 
9 
9 
10 
10 
10 

11 


., 250 

; 227 

A 204 

: iso 

3 157 


674 
644 
614 
584 
554 


42435 
41734 
410 33 
40432 
39731 


10 
10 

10 

11 
11 


2 
3 
3 
3 
3 


30 67866 
31 890 
32 913 
33 936 
34 959 


,32134 
, 110 
* 087 
? 064 
, 041 


73476 
507* 
537^ 
567 1 
597! 
627 
657* 
687, 
717 ^ 
747 


26524 
493 
463 
433 
403 


05610 
617 
624 
631 
638 


9439030 
38329 
376 28 
36927 
362 26 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 


16 
16 
17 
17 
18 
18 
19 
19 
20 
20 


15 
16 
16 

16 
17 
T8 
18 
18 
19 
20 


14 
15 
15 
16 
16 
17 
17 
18 
18 
19 


12 
12 
12 
13 
13 


11 

11 

12 
12 
12 
13 

13 
14 
14 

14 


3 

3 
3 
3 
3 
4 
4 
4 
4 
4 


35 982 

36 68001) 
37 029 
38 052 
39 075 


A 018 
I 31994 
I 971 
I 948 
I 925 


373 
343 
313 
283 
253 
22 
19 
16 
13 
10 


645 
651 
658 
665 
672 


35525 
349 24 
3422 
335 2 
3282 


13 
14 
14 
15 
15 


40 098 
41 121 
42 144 
43 167 
44 190 


, 902 
, 87 
; 85 
H3 

;:; si 


777 
807 
837? 
867 
897 = 

"927 : 

957: 
987: 
74017: 
047: 


679 
686 
693 
700 
707 
" 714 
721 
727 
734 
741 


321 2 

3141 
307 1 
3001 
2931 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


21 
21 

22 
22 

23 


20 

20 
21 
22 
22 
22 
23 
24 
24 
24 


19 

20 

20 
21 

21 


16 

16 

17 

17 
18 
18 
18 
19 
19 
20 


15 
16 
16 
16 
17 


15 
15 

15 
16 

1( L 

16 
17 
17 
18 
18 


5 
5 
5 
5 
5 

~T 

5 
5 
6 
6 


4 

4 
4 
4 
4 


45 213 

46 237 
47 260 
48 283 
49 305 


M 78 

;* 76 

% 74 
w 71 
69 

;! 67 

64 
62 
" 60 
* 58 


07 
04 
01 

2598 
95 


2861 
2791 
2731 
266 1 
2591 


23 
24 

24 
25 

25 


22 
22 

23 
23 

24 


17 

18 
18 
18 
19 


4 
5 

t 

5 
5 


50 328 

51 351 
52 374 
53 397 
54 420 


077 
107 
137 
166 
196 


92 
89 
86 
83 
80 


748 
755 
762 
769 

776 


2521 
245 
238 
231 
224 


50 

51 
52 
53 
54 


26 

26 
27 

27 
28 


25 
26 
26 

26 

27 


24 

25 
25 
26 
26 


20 

20 
21 

21 
22 


19 

20 
20 

20 
21 


18 
19 
19 
19 
20 


6 
6 

6 
6 
6 


5 

5 
5 

e 


55 443 

56 466 
57 489 
58 512 
59 534 


" 55 
" 53 

51 
48 

23 46 

3144 


226 
256 
286 
316 
345 


77 
74 
71 
68 
65 


783 
790 
797 
804 
811 


217 
210 
203 
196 
189 


55 

56 
57 
58 
59 


28 
29 
29 
30 

30 


28 
28 
28 
29 
30 


27 
27 

28 
28 
29 


22 

22 
23 
23 

24 


21 

21 
22 

22 
23 


20 

21 

21 

21 
22 


6 

7 
7 
7 
7 


6 
6 

1 
1 
6 


6068557 


74375 


2562 


05818 


94182 


60 


31 


30 


29 


24 


23 


22 


7 


6 


, 9. 

I cos 


d 10. 

1' I SCO 


9. 

I cot 


10. 

Han 


10. 

I CSC 


9. 

I sin 


" 


31 


30 


29 

Proi 


24 

)orti< 


2* 

>nal 


22 

Part 


7 

5 


1 


118 


61 

73 



29 C 



TABLE II 



150 C 



/ 


I sin 
0. 


d 

1' 

23 
23 
22 
23 
23 
23 
22 
23 
23 
22 
23 
22 
23 
23 
22 
23 
22 
23 
22 
23 
22 
23 
22 
23 
22 
22 
23 
22 
23 
22 
22 
23 
22 

22 
22 
23 
22 
22 

22 
23 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 
22 

I 

1 


1 CSC 

10. 


/tan 
9. 


d 

r 


I cot 
10. 


Jsec 
10. 


d 
r 


/ COS 

9. 


' 


n 


30 


Pro 
29 


portioi 
23 


lal Pa 
22 


rts 
8 


7 




1 

2 
3 

4 


68557 

580 
603 
625 
648 


31443 
420 
397 
375 
352 


74375 
405 
435 
465 
494 


30 
30 
30 
29 
30 
30 
29 
30 
JO 
30 
29 
30 
30 
29 
30 
30 
29 
30 
29 
30 
29 
30 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 
30 
29 
29 
30 
29 
29 
29 
30 
29 
29 
29 
30 
29 
29 
29 
30 
29 
29 

d" 

1 


45625 
595 
565 
535 
506 


05818 
825 
832 
839 
846 


7 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
8 
7 
7 
7 
7 
7 
7 
7 
7 
7 
7 
8 
7 
7 
7 
7 
7 
7 
7 
8 
7 
7 
7 
7 
7 
8 
7 
7 
7 
7 
8 
7 
7 
7 
8 
7 
7 
7 
7 
8 
7 
7 
8 
7 
7 
7 
8 
7 

"d 

1 


94182 
175 
168 
161 
154 


60 

59 
>8 
>7 
56 




1 

2 
3 

4 





1 

2 
2 





1 

2 




1 
1 
2 





1 
1 

1 






1 









5 

6 
7 
8 
9 


671 
694 
716 
739 
762 


329 
306 

284 
261 
238 


524 
554 
583 
613 
643 


476 
446 
417 
387 
357 


853 
860 
867 
874 
881 


147 
140 
133 
126 
119 


55 

54 
53 
52 
51 


5 

6 
7 

8 
9 


2 
3 
4 
4 

4 


2 
3 
3 
4 

4 


2 

2 
3 
3 

3 


2 

2 
3 
3 

3 




1 

1 
1 
1 
1 


10 

11 
12 
13 
14 


784 
807 
829 
852 
875 


216 
193 
171 
148 
125 


673 
702 
732 
762 
791 


327 

298 
268 
238 
209 


888 
895 
902 
910 
917 


112 
105 
098 
090 
083 
076 
069 
062 
055 
048 


50 

49 
48 
47 
46 


10 

11 
12 
13 
14 


5 
6 
6 

G 
7 


5 

5 
6 

7 


4 

4 
5 
5 
5 


4 
4 
4 
5 
5 


2 
2 
2 


1 
1 

1 

2 
2 


15 

16 
17 
18 
19 


897 
920 
942 
965 
987 


103 
080 
058 
035 
013 


821 
851 
880 
910 
939 


179 
149 
120 
090 
061 
031 
002 
24972 
942 
913 


924 
931 
938 
945 
952 


45 

44 
43 
42 
41 


15 

16 
17 
18 
19 


8 
8 
8 
9 
10 


7 
8 
8 
9 
9 


6 
6 

7 
7 

7 


6 
6 
6 
7 
7 


2 

2 
2 
2 
3 


2 
2 
2 

2 
2 


20 

21 
22 
23 
24 


69010 
032 
055 
077 
100 


30990 
968 
945 
923 
900 


969 
998 
75028 
058 
087 


959 
966 
973 

980 
988 


041 
034 
027 
020 
012 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 


10 

10 

11 

12 
12 


10 
10 

11 
11 

12 


8 
8 
8 
9 
9 


7 
8 
8 
8 
9 


3 
3 
3 

3 
3 


2 
2 
3 
3 
3 


25 

26 
27 
28 
29 


122 
144 
167 
189 
212 


878 
856 
833 
811 
788 


117 
146 
176 
205 
235 


883 
854 
824 
795 
765 


995 
06002 
009 
016 
023 


005 
93998 
991 
984 
977 


35 

34 
33 
32 
31 


25 

26 
27 
28 
29 


12 
13 
14 
14 

14 


12 

13 
13 
14 
14 


10 
10 

10 

11 
11 


9 

10 
10 

10 

11 


3 
3 
4 
4 
4 


3 
3 

3 
3 
3 


30 

31 
32 
33 
34 


69234 
256 
279 
301 
323 


30766 
744 
721 
699 
677 


75264 
294 
323 
353 
382 


24736 
706 
677 
647 
618 


06030 
037 
045 
052 
059 


93970 
963 
955 
948 
941 


30 

29 
28 
27 
26 
25 
24 
23 
22 
21 


30 

31 
32 
33 
34 


15 
16 
16 
16 
17 


14 
15 

15 
16 

16 


12 

12 

12 
13 
13 
13 
14 
14 
15 
15 


11 

11 

12 
12 
12 
13 
13 
14 
14 
14 


4 

4 
4 
4 
5 


4 
4 
4 
4 
4 


35 

36 
37 
38 
39 


345 
368 
390 
412 
434 


655 
632 
610 
588 
566 


411 
441 
470 
500 
529 


589 
559 
530 
500 
471 


06f 
073 
080 

088 
095 


934 
927 
920 
912 
905 


35 

36 
37 
38 
39 


18 
18 
18 
19 
20 


17 

17 
18 
18 
19 


5 
5 
5 
5 

5 

" 5 
5 
6 
6 
6 


4 
4 
4 
4 
5 


40 

41 
42 
43 
44 


456 
479 
501 
523 
545 


544 
521 
499 

477 
455 


558 
588 
617 
647 
676 


442 
412 
383 
353 
324 


102 
109 
116 
124 
131 


898 
891 
884 
876 
869 


20 

19 
18 
17 
16 


40 

41 
42 
43 
44 


20 

20 
21 
22 
22 


19 
20 
20 
21 

21 


15 
16 
16 
1C 
17 


15 
15 
15 
16 
16 


5 
5 
5 
5 

5 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


567 
589 
611 
633 
655 


433 
411 
389 
367 
345 


705 
735 
764 
793 
822 
-862 
881 
910 
939 
969 


295 
265 
236 
207 
178 
148 
119 
090 
061 
031 


138 
145 
153 
160 
167 


862 
855 
847 
840 
833 
" "826 
819 
811 
804 
797 


15 

14 
13 
12 
11 


45 

46 
47 
48 
49 


22 
23 
24 
24 

24 


22 
22 
23 
23 

24 


17 
18 
18 

18 
19 


16 

17 

17 
18 
18 


6 

6 
6 
6 

7 


5 
5 
5 
6 
6 


677 
699 
721 
743 
765 


323 
301 
279 
257 
235 


174 
181 
189 
196 
203 


11 


50 

51 
52 
53 
54 


25 
26 
26 

26 
27 


24 

25 
25 

26 
26 


19 

20 
20 

20 
21 


18 
19 
19 

19 
20 


7 
7 
7 

7 
7 


6 
6 

6 
6 
6 


55 

56 
57 
58 
59 
60 
/ 


787 
809 
831 
,853 
875 


213 
191 
169 
147 
126 


998 
76027 
056 
086 
116 


002 
23973 
944 
914 

885 


211 

218 
225 
232 
240 


789 
782 
775 
768 
760 




55 

56 
57 
58 
59 


28 
28 
28 
29 
30 


27 
27 

28 
28 
29 


21 

21 
22 

22 
23 


20 

21 
21 

21 
22 


7 
7 
8 
8 
8 


6 

7 
7 
7 
7 


69897 


30103 


7614' 


23856 


06247 


93753 
9. 

I sin 


( 


60 


30 


29 


23 


22 


8 


7 


9. 

1 COS 


10. 

I sec 


9. 

I cot 


10. 

Ztan 


10. 

I CSC 


/ 




30 


29 

Pr 


23 

Dportic 


22 

nalP 


8 
arts 


7 


119 


60 

74 







30 C 



TABLE II 



149 C 





I sin d 
9. i 


I CSC 

' 10. 


tan d 
9. i 


I cot 
10. 


2 sec d 
10. i 


Zcos 

A 




// 


30 


1* 
29 


ropor 

28 


hona 
22 


tare 
21 


s 

8 


7 




1 

2 
3 

4 


69897 _ 
919 J 
941 * 
963; 
984 * 


,30103 
081 
059 
2 037 
016 


6144 w 
T73* 
202 * 
231 ; 
261^; 


,23856 
827 
798 
769 
739 


6247 , 
254 I 
262 ^ 
269, 
276 I 


93753 

746 
738 
731 
724 




9 

8 
7 
6 




1 

2 
3 
4 





1 

2 
2 




1 

1 
2 




1 
1 
2 




1 
1 

1 





1 
1 
1 






1 









5 

f 
t 

8 
9 


70006 
028. 
050 1 
072 1 
093; 


,29994 
972 
J 950 
: 928 

2 9 7 


290,, 
319* 
348 

377 2 

406 J 


, 710 
681 
652 
623 
j 594 


283 . 

291 ; 

298 I 
305 1 
313 ! 


717 
709 
702 
695 
687 


55 

54 



i2 
51 


5 

6 

7 
8 
9 


2 
3 
4 
4 
4 


2 
3 
3 
4 

4 


2 

3 

3 
4 
4 


2 

2 
3 
3 

3 


2 
2 
2 
3 
3 


1 
1 

1 
1 

1 




10 

11 

f 

15 

.( 

!8 
19 


115 9 
137* 

159 * 
180; 
202; 
224, 
245; 
267; 
288; 

310 2 

"332, 
363 J 
375; 
396, 

418 2 


o 885 
! 863 

: 841 

1 820 
9 798 


435 9 
4647, 
493; 
522, 
551 1 
""580. 
609^ 
639 J 
668, 

697 2 


, 565 
536 
507 
478 
449 


320 , 
327 I 
335* 
342 ! 
350* 


680 
673 
665 
658 
650 


50 

49 

:8 

6 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


5 
6 
6 

6 

7 


5 

5 
6 
6 
7 


5 
5 

6 
6 
7 
7 
7 
8 
8 
9 


4 
4 
4 
5 
5 
6 
6 
6 
7 
7 
- y 

8 
8 
8 
9 


4 
4 
4 
5 
5 


1 
1 
2 
2 
2 


1 
1 
1 

2 
2 


1 776 
J 755 
T 733 
1 712 
j 690 


, 420 

n 39 1 
361 

o 332 

J 303 


357 f 
364 
372 , 
379 , 
386 f 


643 

. 636 
628 
. 621 

; 6u 


45 

[i 

[] 


8 
8 
8 
9 
10 


7 
8 
8 
9 
9 


5 
6 
6 

6 

7 


2 

2 
2 
2 

3 


2 
2 
2 

2 
2 


2< 

j 
2' 


, 668 
J 647 
7 625 
i 604 
f 582 


725 
754 J 

783 2 

812^ 
841^ 


9 275 
9 246 

Q 217 

Q 188 
159 


394 , 
401 
409 f 
416 
423 


. 606 
599 
591 
584 
577 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 


10 

10 

11 

12 
12 


10 
10 

11 

11 

12 


9 
10 

10 

11 
11 


7 

7 
8 
8 
8 


3 
3 
3 

3 
3 


2 
2 
3 
3 
3 


25 

26 

27 
2? 
2| 


439 
461 2 
482* 
504; 
525^ 
70547 
568 
5fl(H 
611 
633^ 


561 
f 539 
., 518 
7 496 

2 475 


87( 
899, 

928 
957, 

986 2 


o 13 

i 101 

9 072 

X 43 

;; OH 


431 
438 
446 
453 
461 
06468 
475 
483 
490 
498 


569 
562 
554 
547 
; 539 


35 

34 
33 
32 

31 


25 

26 
27 
28 
29 


12 
13 
14 
14 

14 


12 

13 
13 
14 
14 


12 
12 
13 
13 

14 


9 

10 
10 

10 

11 


9 
9 
9 
10 
10 


3 
3 
4 
4 
4 


3 
3 

S 
3 
3 


30 

31 
32 
33 
34 


, 29453 
., 432 
f 410 
i 389 
f 367 


77015, 
044, 
073, 
101 
130 


Q 22985 
I 956 

8 927 

I 899 
] 870 


7 93532 
525 
* 517 
510 
j 502 


30 

2< 
2j 
2' 
26 


30 

31 
32 
33 
34 


15 
16 
16 

10 
17 


14 

15 

15 
16 
16 
17 
17 
18 
is 
19 
19 
20 
20 
21 
21 


14 

14 
15 
15 
16 


11 

11 

12 
12 

12 


10 
11 
11 
12 
12 


4 

4 
4 
4 
5 


4 
4 
4 
4 
4 


35 

36 
37 
38 
39 
40 
4 
42 
43 
44 


654. 
675 i 
697;; 
718 
739^ 

76i ; 

782" 
803" 
824 J 
846 \ 


, 346 
J 325 
7 303 
282 
i 26 


159 
188 
217 
246 
274 


Q 84 
o 812 
, 783 
' 754 

N) 72( 


505 
513 
520 
528 
535 


s 495 

! 487 

8 4 ?? 
7 4/2 

H 465 


25 

24 
23 

2i 
2 

^0 

19 
18 
17 
16 
15 
14 
13 
12 
11 


35 

36 
37 
38 
39 


18 
18 

18 
19 
20 
20" 
20 
21 
22 
22 
22 
23 
24 
24 
24 


16 
17 
17 
IS 
18 
19 
19 
20 
20 
21 


13 

13 
14 
14 

14 


12 
13 
13 
13 
14 
14 
14 
15 
15 
15 
16 
16 
16 
17 
17 


5 
5 
5 
5 
5 


4 
4 
4 
4 
5 


. 23$ 
218 
197 
J 176 
154 
133 
112 

i 09 

f 069 
> 048 


303 
332 
361 
390 
418!; 


o 69 
I 668 
, 639 

8 61 
582 


543 
550 
558 
565 
573 


. 457 

8 45 
J 442 

8 435 
J 427 

8 42 
J 412 

8 40 
7 39 

I 390 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


15 
15 
15 
16 
16 


5 
5 
6 
6 
6 


S 

5 
5 

5 


45 

4b 
47 

48 
49 
50 

5 

5! 
EK 
54 


867 , 
888; 
909; 
931 ; 
952^ 


447.. 
476 
505 
533 1 
562^ 


K, <* 
? 52 
8 49 
1 46 
* 43 


580 
588 
595 
603 
610 


22 
22 
23 
23 

24 


21 

21 
22 
22 
23 


16 
17 
17 
IS 
18 


6 

6 
6 
6 

7 


5 
5 
5 

6 
6 


973 r 
994" 
71015' 
036 ; 
058^ 


M 27 
006 

28985 
i 964 
M 942 


591 r 

619; 

648; 
677; 
706" 


8 40 
38 

9 35 

q 32 

>8 29 


618 
625 
633 
640 
648 


, 38 

8 37 

! 36 

8 36 
8 35 


10 


50 

51 
52 
53 
54 


25 
26 
26 

26 
27 


24 

25 
25 
26 
26 


23 
24 
24 
25 
25 


18 
19 
19 

19 
20 


18 
18 
18 
19 
19 


7 
7 
7 

7 

7 


6 
6 

6 
6 
6 


55 

5f 
57 

58 
5 


079, 

100 ; 

12i: 

142; 
163; 


> 92 
900 

87 
85 

! 83 


734, 
763; 
791 ; 
820 ; 
849; 


> 2 

23 
20 

18 
5 15 


656 
663 
671 
678 
686 


f 34 
8 33 

! 32 
! 32 
! 31 
9330 




55 

56 
57 
58 
59 


28 
28 
28 
29 
30 


27 
27 

28 
28 
29 


26 
26 
27 
27 

28 


20 
21 
21 

21 
22 
"22^ 


19 
20 
20 

20 
21 


7 
7 
8 
8 
8 


6 

7 
7 
7 
7 


6 


71184' 


" 2881 


77877 


2212 


06693 




60 


30 


29 


28 


21 
21 

ilPai 


8 


7 


' 


9. 

1 COS 


d 10. 

1' Ise 


9. 

loot 


d 10. 

i' Z tan 


10. 

I CSC 


d 9. 

i Ism 






30 


29 

I 


28 
^ropo 


22 

rtion 


8 
rts 


7 


120 59 

75 





31 C 



TABLE II 



148 



' 


t sin 
9. 


d 

r 

21 
21 
21 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
21 
21 
21 
20 
21 
21 
21 
20 
21 
21 
21 
20 
21 
21 
20 
21 
21 
20 
21 
20 
2] 
20 
21 
20 
21 
21 
20 
20 
21 
20 
21 
20 
21 
20 
20 
21 
20 
20 
21 
20 
20 
21 
20 
20 
21 
20 
20 


I CSC 

10. 


I tan 
9. 


d 

r 


/cot 
10. 


I sec 
10. 


d 
l' 


I COS 

9. 


' 




n 


29 


Pro 

28 


portioi 
21 


lal Pa 
20 


ris 

8 


7 




1 

e 


4 


71184 
205 
226 
247 
268 


28816 
795 
774 
753 
732 


77877 
906 
935 
963 
992 


29 
23 
28 
29 
28 
29 
28 
29 
29 
28 
29 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
29 
28 
'28 
29 
28 
28 
29 
28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
29 
28 
28 
28 
28 
29 
28 
28 

: 

28 
29 
28 
28 
28 


22123 
094 
065 
037 
008 


06693 
701 
709 
716 
724 


8 
8 
7 
8 
7 

8 
8 
7 
8 
8 

8 
8 

I 

8 
7 

8 
8 
7 
8 
8 
7 
8 
8 
7 
8 
8 
8 
7 
8 
8 
8 
7 
8 
8 
8 
8 
7 
8 
8 
8 
7 
8 
8 
8 
8 
8 
7 
8 
8 
8 
8 
8 
8 
7 
8 
8 
8 
8 

I 
1' 


93307 
299 
291 
284 
276 


60 

59 
58 
57 
56 






1 

2 
3 
4 




1 
1 
2 




1 
1 
2 


.0 

1 

1 
1 





1 
1 

1 

2 

3 
3 






1 









5 

6 

7 
8 
c 


289 
310 
331 
352 
373 


711 
690 
669 
648 
627 


78020 
049 
077 
106 
135 


21980 
951 
923 
894 
865 


731 
739 
747 
754 
762 


260 
261 
253 
246 
238 


55 

54 
53 
52 
51 


5 

6 

7 
8 
9 


2 
3 
3 
4 

4 


2 

3 

3 
4 
4 


2 

2 

2 
3 
3 


1 
1 
1 
1 
1 


1 
1 
1 
1 
1 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


393 
414 
435 
456 
477 


607 
586 
565 
544 
523 
~502 
481 
461 
440 
419 


163 
192 
220 
249 
277 
"306 
334 
363 
391 
419 


837 
808 
780 
751 
723 


770 
777 
785 
793 
800 


230 
223 
215 
207 
200 


50 

49 
48 
47 
46 




10 

11 
12 
13 
14 


5 

5 
6 
6 
7 


5 
5 

6 


4 
4 
4 


3 
4 
4 


1 
1 
2 


1 
1 
1 


7 


5 


5 


2 

2 
o 
2 
3 
3 
3 
3 
3 
3 


2 


498 
519 
539 
560 
581 
602 
622 
643 
664 
685 


694 
666 
637 
t}09 
581 
552 
524 
495 
467 
438 
410 
382 
353 
325 
296 
21268 
240 
211 
183 
155 


808 
816 
823 
831 
839 


192 
184 
177 
169 
161 
154 
146 
138 
131 
123 
115 
108 
100 
092 
084 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 


15 

16 
17 
18 
19 


7 
8 
8 
9 
9 


7 

7 
8 

8 

9 


5 
(i 
6 


7 
7 

7 

8 
H 

N 


5 

5 
fi 
6 
(i 

7 
7 

7 
8 
8 


2 
2 
2 

2 

2 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


398 
378 
357 
336 
315 
295 
274 
253 
233 
212 


448 
476 
505 
533 
562 


846 

854 
862 
869 
877 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


10 
10 
11 
11 

12 


9 
10 

10 
11 
11 


2 
2 
3 
3 
3 
3~~ 
3 
3 
3 
3 


705 
726 
747 
767 
788 
71809 
829 
850 
870 
891 


590 
618 
647 
675 
704 


885 
892 
900 
908 
916 


12 

13 
13 

14 
14 

'"l4"" 
15 
15 
16 

10 


12 
12 
13 
13 

14 



9 
9 
10 
10 

"10 
11 

11 

12 
12 


8 
9 
9 
9 
10 
10 
10 
11 
11 
11 


3 
3 
4 
4 
4 


28191 
171 
150 
130 
109 


78732 
760 
789 
817 
845 


06923 
931 
939 

947 
954 


93077 

069 
061 
053 
046 
038 
030 
022 
014 
007 
92999 
991 
983 
976 
968 


30 

29 
28 
27 
20 


30 

31 
32 
33 
34 


14 

14 
15 
15 
16 
~16 
17 
17 
18 
18 


4 

t 
4 
1 
5 
5" 
5 
5 
5 
5 


4 
4 
4 
4 
4 


911 
932 
952 
973 
994 


089 
068 
048 
027 
006 
27986 
966 
945 
925 
904 


874 
902 
930 
959 

987 


126 
098 
070 
041 
013 


9(52 
970 
978 
986 
993 


25 

24 
23 
22 
21 


35 

3(5 
37 
38 
39 


17 

17 
18 
IS 
19 


12 
13 
13 
13 
14 
14 
14 
15 
15 
15 
10 
16 
16 
17 
17 

"is 
18 
18 
19 
19 


12 
12 
12 
U 
13 


4 
4 
4 
4 

(T 

5 " 
5 
5 
5 

5 
5 
5 
5 

(> 
(> 


40 

41 
42 
43 
44 


72014 
034 
055 
075 
096 


79015 
043 
072 
100 
128 


20985 
957 
928 
900 

872 


07001 
009 
017 
024 
032 


20 

19 
18 
17 
16 
I* 
14 
13 
12 
11 


40 

41 
42 
43 
44 
4a 
4(5 
47 
48 
49 


19 
20 
20 
21 

21 


19 
19 
20 
20 
21 
21 
21 
22 
22 
23 


13 
li 
14 
14 
15 
15 
15 
lt> 
16 
Hi 


5 
5 
(i 
6 

6 
G 
6 
6 
7 
7 " 
7 
7 
7 
7 


45 

46 
47 
48 
49 


116 
137 
157 
177 
198 


884 
863 
843 
823 
802 


156 
185 
213 
241 
269 


844 
815 
787 
759 
731 


040 
048 
056 
064 
071 


960 
952 
944 
936 
929 
"921 
913 
905 
897 
889 


22 

22 

23 
23 

24 


50 

51 
52 
53 
54 


218 
238 
259 
279 
299 


782 
762 
741 
721 
701 


297 
326 
354 
382 
410 


703 
674 
646 
618 
590 


079 
087 
095 
103 
111 


10 

r 

J 
i 

(5 


50 

51 
52 
53 
54 


24 

25 
25 
26 
26 


23 
24 
24 
25 
25 


17 
17 

17 
18 
18 


(I 
6 

li 
(> 
6 
G 

7 

7 
7 
7 


55 

56 
57 
58 
59 


320 
340 
360 
381 
401 


680 
660 
640 
619 
599 


438 
466 
495 
523 
551 


562 
534 
505 
477 
449 


119 
126 
134 
142 
150 


881 
874 
866 
858 
850 


5 

4 

1 
1 


55 

5(5 
57 
58 
59 


27 
27 

28 
28 
29 


26 
26 
27 
27 

28 


19 
20 
20 

20 

21 


18 
19 
19 

19 
20 


7 
7 
8 
8 
8 


60 


72421 


27579 


79579 


20421 


07158 


92842 





60 


29 


28 


21 


20 


8 

""" 8" 
rts 


7 
7 


' 


9. 

/ cos 


d 

r 


10. 

I sec 


9. 

I cot 


d 
1' 


10. 

Han 


10. 

I CSC 


9. 

1 sin 


n 


29 


28 
Pro 


21 

portio 


20 

nalPa 


121 58 

76 



32 C 



TABLE II 



147 C 



' 


/ sin 
9. 


d 

20 
20 
21 
20 
20 
20 
20 
20 
20 
20 
21 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
19 
20 
20 
20 
20 
20 
20 
19 
20 
20 
20 
20 
19 
20 
20 
20 
19 
2( 
20 
19 
20 
20 
19 
20 
20 
19 
20 
19 
20 
19 
20 
19 
20 
19 
20 
19 
20 


/ CSC 

10. 


/ tan 
9. 


d 

r 

28 
28 
28 
28 
28 
28 
29 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 

: 

28 
28 

% 
28 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
28 
27 
28 
28 
27 
28 
28 
27 
28 
28 


I cot 
10. 


I sec 
10. 

07158 
166 
174 
182 
190 


d 

r 

8 
8 
8 
8 
7 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
S 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
8 
9 

8 
8 
8 
8 
8 
8 
8 
8 
9 
8 
8 
8 
8 
8 
8 
9 
8 
8 
8 
8 
8 
9 
8 


I COS 

9. 


0(1 

51 
58 
57 
56 


// 

"T 

1 

2 
3 
4 


29 



1 
1 
2 


28 
(Y 


1 

1 

2 


trc 
27 




1 

1 

2 


port 
21 



1 

1 
1 


iona 
20 




1 
1 

1 


1 t>ai 
19 


rts 
9 


8 


7 




1 

2 
3 

4 


72421 
441 
461 
482 
502 
522 
542 
562 
582 
602 


27579 
559 
539 
518 
498 


79579 
607 
635 
663 
691 


0421 
393 
365 
337 
309 


92842 
834 
826 
818 
810 




1 

1 
1 







1 






1 









5 

6 

7 
8 
9 


478 
458 
438 
418 
398 


719 
747 
776 
804 
832 


281 
253 
224 
196 
168 


197 
205 
213 
221 
229 


803 
795 

787 
779 
771 


to 

54 
53 
52 
51 


5 

6 
7 
8 
9 
10 
11 
12 
L3 
H 

Is" 

16 
17 
18 
19 


2 
3 
3 
4 

4 


2 
3 
3 
4 
4 


2 
3 
3 
4 
4 


2 
2 
2 
3 
3 


2 
2 
2 
3 
3 


2 
2 

2 
3 
3 

" 3 
3 
4 


1 
1 
1 
1 
1 


1 
1 

1 
1 
1 

r 

i 

2 


1 


10 

11 
12 
13 
14 


622 
643 
663 
683 
703 


378 
357 
337 
317 
297 
277 
257 
237 
217 
197 


860 
888 
916 
944 
972 


140 
112 
084 
056 
028 


237 
245 
253 
261 
269 
277 
285 
293 
301 
309 


763 
755 
747 
739 
731 
723 
715 
707 
699 
691 


50 

49 
48 
47 
46 


5 

5 


6 
7 
7 
8 
8 
9 
9 
10 
10 
11 
11 
12 


5 
5 

6 


4 
5 

5 


4 

4 

4 


8 
4 
4 


2 
2 
2 


7 
7 

7 
8 
8 
9 





5 

, 

6 
6 

6 

7 


5 


4 

T 
5 

5 

6 


2 
2 
2 
3 
3 
3 
3 
3 
3 
3 
4 
4 
4 
4 
4 
4 
~4 
5 
5 
5 
5 
5 
5 
6 
6 
6 


2 
2~ 

2 
2 
2 
3 


2 
2 
2 
2 

2 
2 


15 

16 
17 
18 
19 


723 
743 
763 
783 
803 
823 
843 
863 
883 
902 
922 
942 
962 
982 
73002 


80000 
028 
056 
084 
112 


000 
19972 
944 
916 

888 


45 

44 
43 
42 
41 


7 
7 

8 
8 
9 


5 

5 
6 
6 

6 


<0 

21 
22 
23 
24 
25 
20 
27 
28 
20 
*G 
31 
,52 
33 
34 
.so 
36 
37 
38 
39 
40 
41 
42 
43 
44 


177 
157 
137 
117 
098 
078 
058 
038 
018 
20998 


140 
168 
11)5 
223 
251 
" 279 
307 
335 
363 
391 


860 
832 
805 
777 
749 


317 
325 
333 
341 
349 


683 
675 
667 
659 
651 


40 

39 
38 
37 
36 
35 
34 
33 
32 
31 


20 

21 
22 
23 
24 


9 
10 

10 

11 
11 


9 

9 
10 

10 

11 

11 

12 
12 
13 
13 
14 
14 
14 
15 
15 


7 

7 
8 
8 
8 
"if 
9 
9 
10 
10 
10 

11 

11 

12 
12 

12 
13 
13 

13 
14 


7 
7 

7 
8 
8 


6 
7 
7 

7 
8 


3 
3 
3 

3 
3 


2 
2 
3 
3 
3 


721 

693 
665 
637 
609 


357 
365 
373 
381 
389 
07397 
405 
413 
421 
429 


643 
635 
627 
619 
611 
92603 
595 
587 
579 
571 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


12 

13 
13 
14 
14 


12 
12 
13 
13 

14 


8 
9 
9 

9 
10 

To 

10 

11 
11 

11 

12 
12 
12 
13 
13 


8 
8 
9 
9 

9 

~io" 

10 

10 

10 

11 


3 
3 
4 
4 
4 
4 
4 
4 
4 

5 
. 

5 
5 
5 

5 


3 
3 

3 
3 

3 
.._. 

4 
4 
4 
4 
4 
4 
4 
4 
5 
5 
5 
5 
5 
5 
5 
5 
5 
6 
6 


73022 
041 
061 

081 
101 


20978 
959 
939 
919 
899 
"879 
860 
840 
820 
800 
781 
761 
741 
722 
702 


80419 
447 
474 
502 
530 


19581 
553 
526 
498 
470 
442 
414 
386 
358 
331 


30 

29 
28 
27 
26 
45 
24 
23 
22 
21 


14 
15 

15 
16 

Ifj 


14 

14 
15 
15 
16 

10 
17 
17 
18 
18 


121 
140 
160 
180 
200 
219 
239 
259 
278 
298 


558 
586 
614 
642 
669 


437 
445 
454 
462 
470 


563 
555 
546 

538 
530 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


17 

17 
18 
18 
19 


16 
16 
17 
17 

18 


11 

11 

12 
12 

12 
13" 

13 

13 
14 

14 

14 
15 
15 
15 
16 
16 
16 
16 
17 
17 


(597 
725 
753 
781 
808 
836 
864 
892 
919 
947 


303 
275 
247 
219 
192 
"164 
13f 
108 
081 
053 
025 
18997 
97( 
942 
914 


478 
486 
494 
502 
510 


522 

514 
506 
498 
490 


20 

19 
18 
17 
16 
15 
14 
13 
12 
11 


19 
20 
20 
21 

21 


19 
19 

20 

20 

21 


18 
18 
19 
19 
20 
20 
21 
21 
22 
22 
22 
23 
23 
24 
24 


14 

14 
15 
15 
15 
10 
10 
16 
17 
17 

18 

18 

18 
19 
19 


13 
14 
14 
14 
15 
U 
15 
16 
10 
16 
17 
17 
17 
18 
18 
18~ 
19 
19 
19 
20 


6 

6 
6 
6 

7 
~J 
7 
7 
7 
7 
8 
8 
8 
8 
8 


5 
5 
6 
6 
6 
6 
6 
6 
6 
7 


45 

46 
47 
48 
49 
oO 
51 
52 
53 
54 


318 
337 
357 
377 
396 


682 
663 
643 
623 
604 
584 
565 
545 
526 
50( 


518 
527 
535 
543 
551 
"559 
567 
575 
584 
592 


482 
473 
465 
457 
449 


22 
22 
23 
23 

24 


21 

21 
22 
22 
23 


416 
435 
455 
474 
494 


975 
81003 
030 
058 
086 
113 
141 
169 
196 
224 


441 
433 
425 
416 
408 


11 

( 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


24 

25 
25 
26 
26 


23 
24 

24 
25 
25 


7 
7 
7 

7 
7 
7 
7 
8 
8 
8 


6 
6 

6 
6 
6 
6 

7 
7 
7 
7 


55 

56 
57 
58 
59 


513 
533 
552 
572 
591 


487 
467 
448 
428 
409 


887 
859 
831 
804 
776 


600 
608 
616 
624 
633 


400 
392 
384 
376 
367 


I 


27 
27 

28 
28 
29 


26 
26 
27 
27 

28 


25 
25 

26 
26 

27 


19 
20 
20 
20 
21 
2f 

^T 

opoi 


17 
18 
18 

18 
19 


8 
8 
9 
9 
9 

~r 


00 


73611 


20389 


81252 


18748 


07641 


92359 


1 


60 


29 


28 


27 


20 


19 


8 


7 


' 


9. 

1 COS 


d 

r 


10. 

I sec 


9. 

I cot 


d 

1 


10. 

Ztan 


10. 

I esc 


d 
1' 


9. 

/ sin 


' 


// 


29 


28 


27 

P 


20 

rtion 


19 

alP 


9 

arts 


8 


7 


122 


57 

77 





33 C 



TABLE II 



146 C 



' 


2 sin 
9. 


d 
1' 


I CSC 

10. 

36389 
370 
350 
331 
311 


I tan 
9. 


d 
i' 


I cot 
10. 


/ sec 
10. 


d 

i 


I COS 

9. 


' 





38 


1^ 
37 


T 


tiona 
19 ] 


I Part 
18 


s 
9 


8 




1 

2 
3 

4 


73611 
630 
650 
669 
689 


19 
20 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
19 
19 
20 
19 
19 
19 
19 
20 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 

in 


81252 
279 
307 
335 
362 


27 
28 
28 
27 
28 
28 
27 
28 
27 
28 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
27 
28 
27 
27 
28 
27 
27 
27 
28 
27 
27 
27 
28 


18748 
721 
693 
665 
638 


07641 
649 
657 
665 
674 


8 
8 
8 
9 
8 
8 
8 
9 
8 
8 
8 
9 
8 
8 
9 
8 
8 
8 
9 
8 
8 
9 
8 
8 
9 
8 
8 
9 
8 
8 
9 
8 
8 
9 
8 
9 
8 
8 
9 
8 
9 
8 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 
8 
9 
8 
cj 
8 
9 
8 
9 


93359 
351 
343 
335 
326 


BO 

59 
58 
57 
56 




1 

2 
3 

4 




1 

1 
3 




1 
1 
3 




1 

1 
1 





1 
1 

1 





1 

1 
1 







1 







1 


& 

6 

7 
8 
9 


708 

727 
747 
766 
785 


292 
273 
253 
234 
215 


390 
418 
445 
473 
500 


610 
582 
555 
527 
500 


682 
690 
698 
707 
715 


318 
310 
302 
293 

285 


55 

'A 

)2 

52 

51 


5 

6 
7 
8 
9 


2 
3 
3 
4 
4 


2 
3 
3 
4 
4 


2 
3 

2 
3 
3 


2 
3 
2 
3 
3 


2 
2 
3 

2 
3 


1 
1 
1 
1 
1 


1 
1 
1 

1 

1 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
30 
21 
22 
25 

24 
35 

26 
27 
28 
29 


805 
824 
843 
863 
882 


195 
176 
157 
137 
118 


528 
556 
583 
611 
638 


472 
444 
417 
389 
362 


723 
731 
740 
748 
756 


277 
269 
260 
252 
244 


50 

49 

48 
47 
46 


10 

11 
12 
13 
14 


5 
5 

6 
6 

7 


4 
5 

5 
6 

6 


3 
4 
4 

4 
5 


3 
3 
4 
4 
4 
5 
5 
5 
6 
6 


3 

3 
4 
4 
4 
4 
5 
5 
5 
6 


2 
2 
2 
3 
2 
2 
2 
3 
3 
3 


1 
1 

2 
2 
2 


901 
921 
940 
959 
978 


099 
079 
060 
041 
022 


666 
693 
721 

748 
776 


334 
307 
279 
252 
224 


765 
773 

781 
789 
798 


235 
227 
219 
211 
202 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 


15 

16 
17 
18 
19 


7 

7 
8 
8 
9 


7 
7 

8 
8 
9 
~ 9 
9 
10 
10 

11 


5 

5 
6 
6 

6 


3 

2 
2 
2 
3 


997 
74017 
036 
055 
074 


003 

35983 
964 
945 
926 


803 
831 
858 
886 
913 
941 
968 
996 
83023 
051 


197 
169 
142 
114 
087 


806 
814 
823 
831 
839 


194 
186 
177 
169 
161 


30 

21 
22 
23 
24 


9 
10 

10 

11 
11 


7 
7 

7 
8 
8 
~8~ 
9 
9 
9 
10 


6 
7 
7 

7 

s 


6 

(i 
7 
7 
7 


3 

3 
3 
3 
4 
4 
4 
4 
4 
4 


3 
3 
3 

3 
3 


093 
113 
132 
151 
170 


907 
887 
868 
849 
830 


059 
032 
004 
17977 
949 


848 
856 
864 
873 
881 


152 
144 
136 
127 
119 


35 

34 
33 
32 
31 


35 

26 
27 
28 
29 


12 
13 
13 
13 
14 
14 
14 
15 
15 
16 


11 

12 
13 
13 
13 
14 
14 
14 
15 
15 


8 
8 
9 
9 

9 


8 
8 
8 
8 
9 
9 
9 
10 
10 
10 


3 
3 
4 
4 
4 


30 

31 
32 
33 
34 


74189 
208 
227 
246 
265 


35811 
792 
773 
754 
735 


83078 
106 
133 
161 

188 


17922 
894 
867 
839 
812 


07889 
898 
906 
914 
923 


93111 
102 
094 
086 
077 


30 

29 

28 
2' 

26 


30 

31 
32 
33 
34 


10 

10 

11 
11 

11 

~12~ 
13 
12 
13 
13 
13 
14 
14 
14 
15 


10 
10 
10 

10 

11 


4 
5 
5 
5 

5 
5~ 
5 

(> 
6 
6 
6 

6 
ft 
G 
7 
7" 
7 
7 
7 
7 


4 

4 
4 
4 
5 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


284 
303 
322 
341 
360 


716 
697 
678 
659 
640 


215 
243 
270 
298 
325 


785 
757 
730 
702 
675 


931 
940 
948 
956 
965 
"973 
982 
990 
998 
08007 


069 
060 
052 
044 
035 


3 
24 
23 
22 
2 
3l 
19 
18 
17 
16 
15 
14 
13 
12 
11 


35 

36 
37 
38 
39 


16 
17 

17 
IS 
18 


16 
16 
17 
17 

18 


11 

11 

12 
13 

12 


10 

11 
11 

11 

12 


5 
5 
5 
5 
5 


379 
398 
417 
436 
455 
474 
493 
512 
531 
549 


621 
602 
583 
564 
545 
526 
507 
488 
469 
451 


352 
380 
407 
435 

462 


648 
620 
593 
565 
538 


027 
018 
010 
002 
91993 


40 

41 
42 
43 
44 


19 
19 
20 
30 

21 


18 
18 
19 
19 
30 
20 
21 
31 
22 
33 


13 
13 
13 
14 
14 


13 

12 
13 
13 

13 


5 
5 

G 

G 


19 
19 
19 
18 
19 
19 
19 
19 
19 
18 
19 
19 
19 
18 
19 

~d 


489 
517 
544 
571 
599 


511 
483 
456 
429 
401 


015 
024 
032 
041 
04< 


985 
976 
968 
959 
951 


45 

46 
47 

48 
49 


31 

21 
33 
22 
33 


15 

15 
16 
16 

16 


14 
15 
15 

15 
16 


14 
14 
14 

14 
15 


6 

6 
6 
6 

7 


50 

51 
52 
53 
54 


568 
587 
606 
625 
644 


432 
413 
394 
375 
356 


626 
653 
681 
708 
735 


374 
347 
319 
292 
265 


058 
066 
075 
083 
092 


942 
934 
925 
917 
908 


10 


50 

51 
52 
53 
54 


23 
34 
24 
25 
35 


22 
33 

23 
34 

24 


17 
17 
17 
18 
18 


16 
16 
16 
17 
17 


15 

15 
16 
16 

16 


8 
8 
8 
8 
8 


7 
7 
7 
7 
7 


56 

56 
57 
58 
59 
60 


662 
681 
700 
719 
737 


338 
319 
300 
281 
263 


762 
790 
817 
844 
871 


238 
210 
183 
156 
129 


100 
109 
117 
126 
134 
08143 


900 
891 
883 
874 
866 




55 

56 
57 
58 
69 


26 
36 
27 
37 

28 


25 
35 

26 
36 

27 


18 
19 
19 

19 
20 


17 
18 
18 
18 
19 


16 
17 
17 

17 
18 


8 
8 
9 
9 
9 


7 
7 
8 
8 
8 


74756 


35244 


83899 


17101 


91857 





60 


38 


37 


30 


19 


18 


9 


8 


' 


9. 

/(TS 


10. 

I sec 


9. 

Zcot 


d 
1 


10. 

Han 


10. 

1 CSC 


d 
1 


9. 

I sin 


f 




38 


37 


30 

ropo 


19 

rtioni 


18 
ilPai 


9 

ts 


8 


123 56 

78 





34 C 



TABLE II 



145 C 





I sin d 
9. l 


/ CSC 

' 10. 


tan d 
9. i 


1 cot 
10. 


sec 
10. 


I COS / 

9. 


a 


28 


1>I 
27 


oporl 
26 1 


,ional 
19 


Dart 
18 


s 
9 


8 




1 

2 
3 

4 


4756, 
775 
794 
812 
831 1 


.25244 
225 
1 206 
188 
I 169 


2899 ,, 
926,, 
9531 
980, 
SOOSg 


. 17101 
074 
047 
020 
16992 


8143 
151 
160 
168 
177 


91857 60 
84959 
84058 
83257 
82356 




1 
2 
3 
4 




1 
1 
2 




1 
1 
2 




1 

1 
2 




1 

1 

1 





1 
1 

1 







1 






1 


5 

6 

7 
8 
c 


850. 
868 
887 J 
906 
924 


I 150 

Q 132 

1 113 
I 094 
9 O? 6 


035 
062, 
089, 
117* 
144^ 


y 965 
938 
[ 911 
! 883 
L 856 


185 
194 
202 
211 
219 


81555 
80654 
79853 
78952 
78151 


5 

6 
7 
8 
9 


2 
3 

3 
4 
4 


2 
3 
3 
4 
4 


2 

3 
3 
3 
4 


2 
2 
2 
3 
3 


2 
2 
2 

2 
3 


1 
1 

1 

1 
1 


1 
1 
1 

1 

1 


10 

11 

e 

i 

A 


943, 
961 
980 
999; 
7501 7 J 


a 057 
1 039 
' 020 

8 (W1 
* 24983 


171 
198 
225, 
252 \ 
280 g 


829 
, 802 
\ 775 

B 748 
* 720 


228 
237 
245 
254 
262 


77250 

76349 
75548 
74647 
738 46 


10 

11 
12 
13 
14 


5 
5 

6 
6 

7 


4 
5 
5 
6 

6 


4 
5 
5 
6 
6 


3 
3 
4 
4 

4 


3 

3 
4 
4 

4 


2 
2 
2 
2 
2 


1 
1 
2 
2 
2 


15 

.6 

8 
19 


036, 
054 1 
073 
091 

110} 


8 964 

Q 946 

8 927 
909 

I 890 


307, 
334 i 
361, 
388, 

415 2 


- 693 
1 666 
i 639 
1 612 
I 585 


271 
280 
288 
297 
305 


72945 
72044 
71243 
70342 
69541 


15 

16 
17 
18 
19 


7 

7 
8 
8 
9 


7 
7 

8 
8 
9 


6 
7 

7 
8 
8 


5 
5 

5 
6 
6 


4 
5 
5 

5 

6 


2 
2 
3 
3 
3 


2 

2 
2 
2 
3 


20 

21 
jf 

2/ 


128 
147 
165 
184 
202 


9 872 
a 853 

* 835 

R 816 
5 798 


442, 
470* 
497, 
524, 
551 p 


8 558 

* 530 
, 503 
, 476 
I 449 


314 
323 
331 
340 
349 


686 40 
67739 
66938 
66037 
65136 


20 

21 
22 
23 
24 


9 
10 
10 
11 
11 


9 

9 
10 

10 

11 


9 
9 
10 
10 

10 


6 
7 
7 

7 
8 


6 

6 
7 
7 

7 


3 

3 
3 
3 
4 


3 
3 
3 
3 
3 


25 

2f 

28 
29 


221 
239 
258 
276 
294 
75313 
331 
350 
368 
386 


8 
9 761 
8 ?42 

! 724 

9 70G 


578, 
605, 
632* 
659, 
6865 


7 422 
- 395 
1 368 
, 341 
7 314 


357 
366 
375 
383 
392 
08401 
409 
418 
427 
435 


64335 
63434 
62533 
61732 
60831 
9159930 
59129 
58228 
57327 
56526 


25 

26 
27 
28 
29 
~ 30 
31 
32 
33 
34 


12 
12 
13 
13 
14 
14 
14 
15 
15 
16 
16 
17 
17 
18 
18 


11 

12 
12 
13 
13 


11 

11 

12 
12 

13 


8 
8 
9 
9 
9 


8 
8 
8 
8 
9 
9 
9 
10 
10 
10 


4 
4 
4 
4 
4 
4 
5 
5 
5 
5 


3 
3 
4 
4 
4 


30 

31 
32 
33 
34 


a 24687 
* 669 
*>50 
! 632 
o 614 


83713 , 
740* 
768, 
795^ 
822^ 


7 16287 

1 260 
! 232 
I 205 

' 178 


14 
14 
14 
15 
15 
16 
16 
17 
17 
18 


13 

13 
14 

14 
15 


10 
10 
10 

10 

11 


4 

4 
4 
4 
5 


35 

36 
37 
38 
39 


405 
423 
441 
459 

478 


* 595 
! 577 
s 559 

n 541 

8 522 


849 2 
876 
903^ 
930; 
957 f 


7 151 
I 124 
I 097 
I 070 
J 043 


444 
453 
462 
470 
479 


55625 
54724 
53823 
1 53022 
5212 


35 

36 
37 
38 
39 


15 

16 
16 
16 
17 


11 

11 

12 
12 

12 


10 

11 
11 

11 

12 


5 
5 
6 
6 
6 


5 
5 
5 
5 
5 


40 

41 
42 
43 
44 
45 
4f 
47 
48 
49 


496 
514 
533 
551 
569 


Q 504 
o 48b 
a 467 

a 449 

8 431 


984 
84011' 
038' 
065 ' 
092^ 


7 016 
1 15989 
I 962 
I 935 
J 908 


488 
496 
505 
514 
523 


1 5122 

D 5041 

9 4951 
? 4861 

1 4771 


40 

41 
42 
43 
44 


19 
19 
20 
20 
21 
21 
21 
22 
22 
23 


18 

18 
19 
19 
20 


17 
18 
18 
19 
19 


13 
13 

13 
14 
14 


12 

12 
13 
13 
13 


6 

6 
6 
6 
7 
7 
7 
7 
7 
7 


5 
5 
6 
6 
6 


587 
605 
624 
642 
660 


413 
o 395 
376 
358 

8 34 


119. 
146 : 

173: 

200; 

227: 


> 7 88 
! 854 
! 827 

;; soo 

1 773 


531 
540 
549 
558 
567 


, 4 691 
? 4601 

9 45U 

9 4421 
J 4331 


45 

46 
47 

48 
49 


20 
21 
21 
22 
22 


20 
20 
20 
21 

21 


14 
15 
15 

15 
1C 


14 
14 
14 

14 
15 


6 

6 
6 
6 

7 


50 
5 

5! 
5< 
54 


678 
696 
714 
733 
751 


322 

8 304 

! 28f 

a 267 

8 249 


254 . 
280' 
307; 
334 A 
361 ' 


>6 74 

5 720 
693 
! 66 
" 63 


575 
584 
593 
602 
611 


S5' 

is 

8 389 


50 

51 
52 
53 
54 


23 
24 
24 
25 
25 


22 
23 

23 
24 

24 


22 
22 
23 
23 

23 


16 
16 

16 
17 
17 


15 

15 
16 
16 

16 


8 
8 
8 
8 
8 


7 
7 
7 
7 

7 


5 

56 

5' 
5 
5<) 


769 
787 
805 
823 
841 


o 23 
! 213 
! 195 
! 177 
8 159 


388 
415 ' 
442' 
469 
496 


27 61 

! 58 
7 558 

7 53 

* 50 


619 
628 
637 
646 
655 


: s 
a 

J 345 


55 

56 
57 

58 
59 


26 
26 
27 
27 

28 


25 
25 

26 
26 

27 


24 

24 
25 
25 

26 


17 
18 
18 
18 
19 


16 
17 
17 

17 
18 


8 
8 
9 
9 
9 


7 
7 

8 
8 
8 


60 


75855) 


2414 


84523 


1547 


08664 


91336 


60 


28 


27 


26 
~26^ 

^ropo 


19 
~19 

rtioni 


18 


9 


8 


9. 

I COS 


d 10. 

1 / HOC 


9. 

1 cot 


d 10. 

1' Ztai 


10. 

J CSC 


d 9. 

i' 1 sin 


// 


28 


27 

I 


18 

ilPai 


9 

ts 


8 


124 55 

79 



35 C 



TABLE II 



144 C 



' 


{sin 
9. 


d 
1' 


I CSC 

10. 


Jtan 
9. 


d 
1' 


I cot 
10. 


I sec 
10. 

08664 
672 
681 
690 
699 


d 
1' 

8 

g 

9 
9 
9 
fl 
9 
8 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
8 
9 
9 
9 
9 
i) 
9 
9 


9 
9 
9 
9 
10 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
10 
9 
9 
9 
9 
9 
9 
10 
9 
9 
9 
9 


1 COS 

9. 


' 




n 


27 


P 
26 



1 

1 
2 


ropor 
18 


tiona 
17 


l>art 
10 


s 

9 


8 




] 

i 


75859 
877 
895 
913 
931 


18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
17 
18 
18 
18 
18 
18 
17 
18 
18 
18 
17 
18 
18 
18 
17 
18 
18 
17 
18 
18 
17 
18 
18 
17 
18 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
18 
17 
1H 


94141 
123 
105 
087 
069 


84523 
550 
576 
603 
630 


27 
26 
27 
27 
27 
27 
27 
27 
20 
27 
27 
27 
27 
27 
26 
27 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
27 
26 
27 
27 
2(5 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
27 
26 
27 
26 
27 
27 
26 
27 
26 
27 
27 
26 


15477 
450 
424 
397 
370 


91336 
328 
319 
310 
301 


60 

59 
58 
57 
56 




1 
2 
3 
4 




1 
1 
2 




1 
1 
1 
" 2 
2 
2 
2 
3 





1 
1 
1 







1 







1 
i 
i 
i 

i 
i 






1 


5 

( 

8 
9 


949 
967 
985 
76003 
021 


051 
033 
015 
83997 
979 


657 
684 
711 
738 
764 


343 
316 
289 
262 
236 


708 
717 
726 
734 
743 


292 
283 
274 
266 
257 
'248 
239 
230 
221 
212 


55 

54 
53 
52 
51 
50 
49 
48 
47 
4(> 


5 

6 

7 
8 
9 


2 
3 
3 
4 
4 


2 

3 
3 
3 
4 
4 
5 
5 
G 
6 


1 

2 
2 

2 
3 


1 

1 
1 
1 

2 




10 

11 
12 
13 
14 


039 
057 
075 
093 
111 


961 
943 
925 
907 

889 


791 
818 
845 
872 
899 


209 
182 
155 
128 
101 


752 

761 
770 
779 

788 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


4 
5 
5 
6 

6 


3 

3 
4 
4 
4 
4 
5 
5 
5 
G 


3 
3 

3 
4 
4 


2 
2 
2 

2 
2 


2 
2 
2 
2 

2 


1 
1 
2 
2 
2 


15 

16 
17 
18 
19 


129 
146 
1-64 
182 
200 


871 

854 
836 
818 
800 


925 
952 
979 
85006 
033 


075 
048 
021 
14994 
967 


797 
806 
815 
824 
833 


203 
194 
185 
176 
167 


46 

44 
43 
42 
41 
40 
30 
38 
37 
36 
35 
34 
33 
32 
31 


7 
7 

8 
8 
9 


G 
7 

7 
8 

8 


4 
5 
5 
5 

5 


2 
3 
3 
3 
3 
3 
4 
4 
4 
4 


2 
2 
3 
3 
3 


2 

2 
2 

2 
3 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


218 
236 
253 
271 
289 
307 
324 
342 
360 
378 


782 
764 
747 
729 
711 


059 
086 
113 
140 
166 


941 
914 

887 
860 
834 


842 
851 
859 
868 
877 


158 
149 
141 
132 
123 
114 
105 
096 
087 
078 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


9 

9 
10 

10 

11 

"ll 

12 
12 
13 
13 
14 
14 
14 
15 
15 
10 
16 
17 
17 
18 
18 
IS 

19 

19 
20 
20 
21 
21 
22 
22 


9 
9 
10 
10 

10 

if 

n 

12 
12 
13 
13 
13 
14 
14 
15 
15 
10 
16 
10 
17 
17 
18 
18 
19 
19 
20 
20 
20 
21 
21 


6 

G 
7 
7 

7 


G 
6 

G 

7 
7 


3 

3 
3 
3 

4 
~T 
4 
4 

4 
4 


3 
3 
3 

3 
3 
3 
3 

4 
4 
4 


693 
676 
658 
640 
622 
23605 
587 
569 
552 
534 


193 
220 
247 
273 
300 
85327 
354 
380 
407 
434 


807 
780 
753 
727 
700 


88fi 
895 
904 
913 
922 


8 
8 
8 
8 
9 
9 
9 
10 
10 
10 


7 

7 
8 
8 
8 


4 
4 
4 
5 
5 


76395 
413 
431 
448 
466 


14673 
646 
620 
593 
566 


08931 
940 
949 
958 
967 


91069 
060 
051 
042 
033 


30 

29 
28 
27 
26 


30 

31 
32 
33 
34 


8 
9 
9 
9 
10 
10 
10 
10 

11 
11 

11 

12 
12 

12 
12 


5 

5 
5 

6 
G 
G 
6 

G 
G 
G 


4 
5 
5 
5 

5 

5 
5 
G 
G 
G 
6 
6 
6 
G 
7 
' 1 
7 
7 
7 
7 


4 

4 
4 
4 
5 
5" 
5 
5 
5 
5 


35 

36 
37 
38 
39 


484 
501 
519 
537 
554 


516 
499 
481 
463 
446 


460 
487 
514 
540 
567 


540 
513 
486 
460 
433 


977 
986 
995 
09004 
013 


023 
014 
005 
90996 
987 


26 

24 
23 
22 

21 




35 

36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


10 

11 
11 

11 

12 
12 

12 
13 
13 
13 
14 
14 
14 
14 
15 
15 
15 
10 
16 
1G 


40 

41 
42 
43 
44 


572 
590 
607 
625 
642 


428 
410 
393 
375 
358 
340 
323 
305 
288 
270 


594 
620 
647 
674 
700 
"727 
754 
780 
807 
834 


406 
380 
353 
326 
300 


022 
031 
040 
049 
058 
067 
076 
085 
094 
104 


978 
969 
960 
951 
942 
933 
924 
915 
906 
896 


20 

IS 
18 
17 
Hi 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 


7 
7 
7 

7 

7 


5 
5 

G 
G 
G 


45 

46 
47 
48 
49 


660 
677 
695 
712 
730 


273 
246 
220 
193 
166 


13 
13 
13 
14 
14 


8 
8 
8 
8 

s 

8 
8 
9 
9 
9 
9 
9 
10 
10 
10 


6 

G 
G 
6 

7 


50 

51 
52 
53 
54 


747 
765 
782 
800 
817 


253 
235 
218 
200 
183 


860 
887 
913 
940 
967 


140 
113 
087 
060 
033 


113 
122 
131 
140 
149 


887 
878 
869 
860 
851 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


22 
23 

23 
24 

24 


22 
22 
23 
23 

23 


14 
14 
15 
15 
15 
"10 
16 
IB 
16 
17 
17~ 
17 
tiona 


8 
8 
8 
8 
8 


7 
7 
7 

7 
7 


55 

56 
57 
58 
59 
60 


835 
852 
870 
887 
904 
76922 


17 
18 
17 
17 
18 


165 
148 
130 
113 
096 


993 
86020 
046 
073 
100 
86126 


007 
13980 
954 
927 
900 


158 
168 
177 
186 
195 


842 
832 
823 
814 
805 


5 

4 

g 

2 
1 



25 
25 

2G 
26 

27 


24 

24 
25 
25 

2C 


Hi 
17 
17 
17 
18 
'lT 


8 
8 
9 
9 
9 


7 
7 
8 
8 

8 
._._.. 


23078 


13874 


09204 


90796 


27 


26 


10 
10 

1 Par 


:) 
9 

ts 


' 


9. 

1 COS 


d 

1' 


10. 

/sec 


9. 

I cot I 


d 

1' 


10. 

/tan 


10. 

1 CMC 


d 

r 


9. 

I sin 


/ 






27 


26 
P 


18 
ropoi 


8 


125 54 

80 



36 C 



TABLE II 



143 C 



t 


I sin 


d 


1 CSC 


I tan 


d 


/ cot 


/ sec 


d 


I COS 


t 




Proportional Parts 




9. 


1' 


10. 


9. 


l' 


10. 


10. 


i' 


9. 






27 


26 


18 


17 


16 


10 


9 


(1 


76922 




23078 


86126 




13874 


09204 




9079(5 


60 


























1 


939 




061 


153 




847 


213 




787 


59 


1 























2 


957 




043 


179 




821 


223 


10 


777 


58 


2 


1 


1 


1 


1 


1 








3 


974 




026 


206 




794 


232 


y 


7(58 


57 


3 


1 


1 


1 


1 


1 








4 


991 




009 


232 


97 


768 


241 


y 

9 


759 


56 


4 


2 


2 


1 


1 


1 


1 


1 


5 


77009 




22991 


259 




741 


250 




750 


55 


5 


2 


2 


2 


1 


1 


1 


1 


6 


026 




974 


285 




715 


259 




741 


54 


(5 


3 










1 


1 


7 


043 


n 


957 


312 


11 


688 


2(59 


10 


731 


K 


7 


3 


3 


2 


2 


2 


1 


1 


8 


061 




939 


338 




662 


278 




722 


52 


8 


4 


3 


2 


2 


2 


1 


1 


9 


078 


17 


922 


365 


'J7 


635 


287 




713 


51 


9 


4 


4 


3 


3 


2 


2 


1 


10 


095 




905 


392 




608 


296 




704 


i( 


10 


4 


4 


3 


3 


3 


2 


2 


11 


112 




888 


418 




582 


306 




694 


49 


11 


5 


5 


3 


3 


3 


2 


2 


12 


130 




870 


446 




555 


315 


y 


(585 


48 


12 


5 


5 


4 


3 


3 


2 


2 


13 


147 


u 


853 


471 




529 


324 


y 


676 


47 


13 


6 


G 


4 


4 


3 


2 


2 


14 


164 


17 


836 


498 




502 


333 


10 


6(57 


46 


14 


G 


6 


4 


4 


4 


2 


2 


L5 


181 




819 


524 




47(5 


343 




657 


L5 


15 


7 


G 


4 


4 


4 


2 


2 


16 


199 




801 


551 


*7 


449 


352 




648 


44 


16 


7 


7 


5 


5 


4 


3 


2 


17 


216 




784 


577 




423 


361 




639 


43 


17 


8 


7 


5 


5 


5 


3 


3 


18 


233 


I/ 


767 


603 




397 


370 




630 


42 


18 


8 


8 


5 


5 


5 


3 


3 


19 


250 


iv 

ift 


750 


630 




370 


380 


q 


620 


41 


19 




8 


fi 


5 


5 


3 


3 


2C 


268 




732 


056 




344 


389 




611 


4( 


20 


9 


9 


6 


G 


5 


3 


3 


21 


285 




715 


683 




317 


398 




602 




21 


9 


9 





6 


G 


4 


3 


22 


302 




698 


709 


2C 


291 


408 




592 


38 


22 


10 


10 


7 


6 


6 


4 


3 


23 


319 




681 


736 l26 


264 


417 


^ 


583 


37 


23 


10 


10 


7 


7 


6 


4 


3 


24 


336 


17 


664 


782,2 


238 


426 


;J 


574 


36 


24 


11 


10 


7 


7 


6 


4 


4 




353 




647 


789U 


211 


435 


in' 565 


35 


25 


11 


11 


8 


7 


7 


4 


4 


2(5 


370 




630 


81 5 If 


185 


445 


1 555 


34 


26 


12 


11 


8 


7 


7 


4 


4 


27 


387 




613 


842 ' 


158 


454 




546 


33 


27 


12 


12 


8 


8 


7 


4 


4 




405 


18 


595 


868 i^' 


132 


463 




537 


32 


28 


13 


12 


8 


H 


7 


5 


4 


29 


422 


IV 
17 


578 


894^ 


106 


473 


'" 527 


31 


29 


13 


13 


9 


8 


8 


5 


4 


30 


77439 




22561 


8692 C 


iam 


09482 


,,90518 


30 


30 


14 


13 


9 


ft 


8 


5 


4 


31 


456 




544 


947 : 


053 


491 




509 


29 


31 


14 


13 


9 





8 


5 


5 


32 


473 




527 


974;! 


02(5 


501 *;; 


499 


28 


32 


14 


14 


10 


9 


9 


5 


5 


33 


490 




510 


87000$ 


000 


510' 490 


27 


33 


15 


14 


10 


9 


9 





5 


34 


507 


IV 
17 


493 




12973 


520I 1 "' 480 


26 


34 


15 


15 


10 10 


9 


G 


5 


35 


524 




476 


053 


947 


529| ,. 471 


25 


35 


10 


15 


10 10 


9 


G 


5 


3( 


541 




459 


079!*' 


921 


5381. j 462 


2' 


36 


16 


10 


11 10 


10 


6 


5 


37 


558 




442 


106;! 


894 


548 


o 452 


23 


37 


17 


16 


11 10 


10 


G 


6 


38 


575 




425 


1321* 


868 


557 


! 443 


22 


38 


17 


10 


11 11 


10 


6 


6 


39 


592 


17 


408 


158 




842 


566 


, 434 


21 


39 


18 


17 


12 11 


10 


6 


6 


4( 


(509 




391 


185 




81,5 


57(5 


J 424 


20 


40 


18 


17 


12 


11 


11 


7 


6 


41 


626 




374 


211 




789 


585 


, 4 15 


19 


41 


IS 


18 


12 




11 


7 


6 


42 


643 


i; 


357 


238 




762 


595 


10 


405 


IS 


42 


19 


18 


13 


12 


11 


7 


6 


43 


660 


i; 


340 


264 




73(5 


604 


y 


396 


17 


43 


19 


19 


13 




11 


7 


6 


44 


677 


i; 

17 


323 


290 




710 


614 


10 


38( 




44 


20 


19 


13 


12 


12 


7 


7 


45 


694 




30( 


317 




683 


623 




377 


15 


45 


20 


20 


14 


13 


12 


8 


7 


4f 


711 




289 


343 




6,57 


(532 


y 


368 


14 


46 


21 


20 


14 


13 


12 


S 


7 


47 


728 




272 


369 




631 


642 


10 


3,58 


13 


47 


21 


20 


14 


13 


13 


8 


7 


48 


744 


Ib 


25f 


396 




(504 


651 


y 


349 


12 


48 


22 


21 


14 


14 


13 


8 


7 


49 


761 


I/ 
17 


239 


422 




578 


661 


10 


339 


11 


49 


22 




15 


14 


13 


8 


7 


50 


778 




222 


448 




5,52 


670 




330 


10 


50 


22 


22 


15 


14 


13 


8 


8 


51 


795 




205 


475 




52,5 


680 


10 


32( 




51 


23 


22 


15 


14 


14 


8 


S 


52 


812 




188 


501 




499 


689 




311 




52 


23 


23 


10 


15 


U 


9 


8 


53 


829 


U 


171 


527 




473 


699 


10 


301 




53 


24 


23 


16 


U 


14 





8 


54 


846 


U 


154 


554 




440 


708 


10 


292 




54 


24 


23 


10 


15 


H 


9 


8 


55 


862 




13? 


580 




42( 


71H 




282 




55 


25 


24 


10 


10 


15 


9 


8 


5b 


879 




121 


(506 




394 


727 




273 




56 


25 


24 


17 


16 


15 


9 


8 


57 


896 




104 


633 




367 


737 


10 


263 




57 


20 


25 


17 


10 


15 


10 


9 


58 


913 


[i 


087 


659 


2( 


341 


746 


y 


254 




58 


26 


25 


17 


10 


15 


10 


9 


59 


930 


10 


070 


685 


26 


315 


756 


u 


244 




59 


27 


26 


18 


17 


10 


10 


9 


60 


77946 




22054 


87711 




12289 


097(55 




90235 





60 


27 


26 


18 


17 


16 


10 


9 


, 


9. 


d 


10. 


9. 


d 


10. 


10. 


d 


9. 


f 


" 


27 


26 


18 


17 


16 


10 


9 




1 COM 


r 


I sec 


J oot 


r 


1 tan 


I CSC 


i' 


I sin 






Proportional Parts 


126 


53 




81 



37 C 



TABLE II 



142 C 



/ 


I sin 
9. 


d 
]/ 

17 
17 
17 
16 

17 


/CSC 

10. 


Han 
9. 


d 
1' 


/cot 
10. 


/ sec 
10. 


d 

i' 


/ COS 

9. 


/ 






1 

2 
3 

4 


27 


Pro 
20 


portio 
17 


rial Pa 
16 


rts 
10 


9 




1 

2 
3 
4 


77946 
963 
980 
997 
78013 


29054 
037 
020 
003 

21987 


87711 
738 
764 
790 
817 


27 
26 
26 
27 
26 
26 
26 
27 
26 
26 
26 
27 
26 
26 
26 
26 
27 
26 
26 
26 
26 
27 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
27 
26 
26 
26 
26 
26 
26 
2f 
26 


12289 
262 
236 
210 
183 


09765 
775 
784 
794 
803 


10 
9 
10 
9 
10 


90235 
225 
216 
206 
197 


60 

59 
58 
57 
56 




1 
1 
2 





1 

1 

2 





1 
1 

1 




1 
1 

1 







1 







1 


5 

6 
7 
8 
9 


030 
047 
063 
080 
097 


17 
16 
17 
17 
16 


970 
953 
937 
920 
903 


843 
869 
895 
922 
948 


157 
131 
105 
078 
052 


813 
822 
832 
841 
851 


9 

10 
9 
10 
10 
9 
10 
9 
10 
10 
9 
10 

y 

10 
10 
9 
10 
10 
9 
10 
10 
9 
10 
10 
9 
10 
10 
9 
10 
10 
10 
9 
10 
10 
10 
f 
10 
10 
10 

10 
10 
10 
10 
9 
10 
10 
10 
10 
10 
9 
10 
10 
10 
10 

I 

1' 


187 
178 
168 
159 
149 


55 

54 
53 
52 
51 




5 

(> 
7 
8 
9 


3 
3 
4 
4 


2 

3 
3 
3 
4 


1 

2 
2 

2 
3 


1 
2 
2 
2 
2 


1 

1 
1 
1 

2 


1 
1 

1 

1 
1 


10 

11 
12 
13 
14 


113 
130 
147 
163 
180 


17 
17 
16 
17 
17 
16 
17 
16 
17 
17 
16 
17 
16 
17 
16 
17 
16 
17 
16 
17 
16 
17 
16 
16 
17 
16 
17 
16 
16 
17 
16 
17 
16 
16 
17 
16 
16 
16 
17 
16 
16 
17 
16 
16 
16 
16 

16 
16 
16 

7 
1' 


887 
870 
853 
837 
820 


974 
88000 
027 
053 
079 


026 
000 
11973 
947 
921 


861 
870 
880 
889 
899 


139 
130 
120 
111 
101 


50 

49 

48 
47 
46 




10 

11 
12 
13 
14 


4 
5 
5 
6 

6 


4 
5 
5 

6 


3 
3 

3 
4 
4 


3 
3 

3 
3 
4 


2 
2 
2 

2 
2 


2 
2 
2 
2 

2 


15 

16 
17 
18 
19 


197 
213 
230 
246 
263 


803 
787 
770 
754 
737 


105 
131 
158 
184 
210 


895 
869 
842 
816 
790 


909 
918 
928 
937 
947 


091 
082 
072 
063 
053 
"043 
034 
024 
014 
005 
89995 
985 
976 
9()(> 
956 


45 

44 
43 
42 
41 
40 
39 
38 
37 
3C 




15 

16 
17 
18 
19 


7 
7 

8 
8 
9 


6 
7 

7 
8 

8 


4 
5 
5 
5 

5 


4 

4 
5 
5 
5 


2 
3 
3 
3 

3 


2 
2 
3 
3 
3 


20 

21 
22 
23 

24 


280 
296 
313 
329 
346 


720 
704 
687 
671 
654 


236 
262 
289 
315 
341 


764 
738 
711 
685 
659 


957 
966 
976 
986 
995 




20 

21 
22 
23 
24 


9 

9 
10 

10 

11 


9 
9 
10 
10 

10 


(> 
6 

6 

7 
7 


5 
6 
6 

6 
6 


3 
4 
4 
4 
4 


3 

3 
3 
3 
4 
4 
4 
4 
4 
4 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


362 
379 
395 
412 
428 
78445 
461 
478 
494 
510 


638 
621 
605 
588 
572 


367 
393 
420 
446 
472 


633 
607 
580 
554 

528 


10005 
015 
024 
034 
044 


35 

34 
33 
32 
31 


25 

20 
27 
28 
29 
30 
31 
32 
33 
34 
35 
3t> 
37 
38 
39 


11 

12 
12 
13 
13 


11 

11 

12 
12 
13 
13" 

13 
14 

14 
15 


7 

7 
8 
8 
8 
~~8 
9 
9 
9 
10 


7 
7 

7 
7 
8 


4 
4 
4 
5 
5 
5" 
5 
5 
(i 
6 


21555 
530 
522 
506 
490 


88498 
524 
550 
577 
603 


11502 
476 
450 
423 
397 


10053 
063 
073 
082 
092 


89947 
937 
927 
918 
908 


30 

29 

28 
27 
26 


M 
14 

14 
15 

15 

"Tfl " 
16 
17 
17 

IS 


8 
8 
9 
9 
9 


4 
5 
5 
5 
5 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


527 
543 
560 
576 
592 


473 
457 
440 
424 
408 


629 
655 
681 
707 
733 


371 
345 
319 
293 
267 


102 
112 
121 
131 

141 


898 
888 
879 
869 
859 


25 

24 
23 
22 
21 




15 

16 
16 
16 
17 


10 

10 
10 

11 
11 


9 
10 
10 
10 

10 

if" 
11 

11 
11 

12 


6 
6 

6 
6 
6 

7~ 
7 
7 

7 
7 


5 
5 

6 

6 


609 
625 
642 
658 
674 


391 
375 
358 
342 
326 


759 
786 
812 
838 
864 


241 
214 
188 
162 
136 


151 
160 
170 
180 
190 


849 
840 
830 
82( 
810 
'801 
791 
781 
771 
761 


20 

10 
18 
17 
16 




40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


18 
18 
19 
19 
20 
'20 " 
21 
21 
22 
22 


17 
18 
18 
19 
19 


11 

12 

12 
12 


6 

tt 
6 
G 
7 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


691 
707 
723 
739 
756 


309 
293 
277 
261 
244 


890 
916 
942 
968 
994 


110 
084 
058 
032 
006 


199 
209 
219 
229 
239 


15 

14 
13 
12 
11 




20 
20 
20 
21 

21 


13 
13 

13 
14 
14 


12 

12 
13 
13 
13 
13 
14 
14 
14 
14 


8 
8 
8 
8 
8 
8 
8 
9 
9 
9 


7 
7 
7 

7 
7 
"~8~ 
8 
8 
8 
8 


772 
788 
805 
821 
837 
~~8"53 
869 
886 
902 
918 


228 
212 
195 
179 
163 


89020 
046 
073 
099 
125 


10980 
954 
927 
901 
875 


248 
258 
268 

278 
288 


752 
742 
732 
722 
712 


10 

9 
* 

6 


50 

51 
52 
53 
54 


22 
23 

23 
24 

24 


22 

22 

23 
23 

23 
24 

24 
25 
25 

20 


14 
14 
15 
15 

15 

"Ii>" 
18 
10 
Hi 
17 


55 

56 
57 
58 
59 
00 


147 
131 
114 
098 
082 


151 
177 
203 
229 
255 


849 
823 
797 
771 
745 


298 
307 
317 
327 
337 


702 
693 
683 
673 
663 


5 

~o 




55 

56 
57 
58 
59 
60 


25 
25 
20 
2fi 
27 
27 


15 
15 

15 
15 
Hi 


9 
9 
10 
10 
10 


8 
8 
9 
9 
9 
9 


78934 


21066 


89281 


10719 


10347 


89653 


26 


17 


16 


10 


' 


9. 

1 COS 


10. 

Zsec 


9. 

lent 


d 

1 


10. 

/tan 


10. 

/ CMC 


9. 

/ sin 


' 




27 


26 

Pr 


17 

Dportu 


16 

nalP 


10 

arts 


9 


127 


52 

82 



38 C 



TABLE II 



' 


I sin 
9. 


d 

t 


JCSC 

10. 


Itan d 
9. 1 


I cot 
' 10. 


/ SPC d 
10. i 


1 CUM 

9. 




// 


26 


25 


l>rop 
17 


ortioi 
16 


lalP 
15 


arts 
11 


10 


9 




1 

2 
3 

4 


78934 
950 
967 
983 
999 


6 
7 
6 
6 
6 
16 
6 
6 
6 
fl 


21066 
050 
033 
017 
001 


89281 0< 
307* 
333* 
359^ 
385, 


. 10719 
693 
! 667 
! 641 

; 615 


0347 

357 
367 * 
376 ; 
386 


,89653 
643 
633 
624 
J 614 


BO 

9 
8 
7 

56 




1 

2 
3 
4 





1 

1 

2 
2 
3 
3 
3 
4 




1 

1 
2 





1 
1 

1 




1 
1 
1 






1 

1 












1 







1 


5 

6 

7 
8 
9 


79015 
031 
047 
053 
079 


90985 
969 
953 
937 
921 


411 
437* 
463; 
489, 
515 2 


589 
563 
537 
511 

I 485 


396 
406 
416 
426 
436 < 


, 604 
594 
; 584 
574 
564 


55 

c 

l] 


5 

6 
7 
8 
9 


2 

2 
3 

3 

4 

T 

5 
5 

5 
6 

6~ 
7 
7 
8 
8 
8~" 
9 
9 
10 
10 


1 

2 
2 

2 
3 


1 
2 
2 
2 
2 


1 

2 
2 
2 

2 


1 

2 


1 
1 

1 
1 

2 




10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


095 
111 
128 
144 
160 
"176 
192 
208 
224 
240 


16 
17 
16 
16 
16 
16 
16 
16 
16 
fi 


905 
889 
872 
856 
840 


541 
567^ 
593* 
619^ 

.J&l 
671 
697* 
723* 
749* 
775* 


. 459 
433 
? 407 

* m 
I 355 


446 
456 
466 
476 
486} 


. 554 
: 544 
: 534 
\ 524 
3 514 
, Ii04 
\ 495 
n 485 
J 475 
n 465 


50 

49 

8 

le 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 


4 
5 
5 
6 
_6 
6 
7 
7 
8 
8 


3 
3 

3 
4 
4 


3 
3 

3 
3 
4 


2 
3 
3 
3 
4 
~4 
4 
4 
4 
5 
5 
5 
6 
6 
6 


2 
2 

2 
2 
3 


2 
2 
2 
2 
2 
2 
3 
3 
3 
3 
3 
4 
4 
4 
4 


2 
2 
2 
2 
2 


824 
808 
792 
776 
760 


329 
J 303 
I 277 

R 251 

P 225 
~199 
! 173 
5 147 

121 

r 095 


496 
505 
515 
525 
_535 
545 
555 
565 

585 * 


45 

[C 

ii 
n 


4 
5 
5 
5 
5 


4 

4 
5 
5 
5 


3 
3 

3 
3 
3 


2 
2 
3 
3 
3 
3 
3 
3 
3 
4 


256 
272 
288 
304 
319 


16 
16 
16 

16 
16 
16 
16 
16 
16 
16 
16 
16 
15 
16 
16 
If 
16 
16 
15 
16 
16 
16 
15 
11 

16 
16 
15 
16 
16 
15 
16 
16 
15 
16 
16 
15 
16 
16 
15 

c 

1 


744 
728 
712 
696 
681 


801 
827 1 
853* 
879 1 

905 2 


455 
] 445 
435 
425 

415 


9 
9 
10 
10 

10 


6 
6 

6 

7 
7 


5 
6 
6 

6 
6 


4 
4 
4 

4 

4 


25 

26 
27 
28 
2<: 

30 

31 
32 
33 
34 


335 
351 
367 
383 
399 
79~415 
431 
447 
463 
478 


665 
649 
633 
617 
601 
20585 
569 
553 
537 
522 


931 
957^ 
983 1 
90009^ 
035 g 
90061. 
086; 
112* 

138 2 

164^ 


A 69 

! 043 
! 017 
09991 

6 965 


5951, 
605 
615 l 
625 
636 
10640 , 
656 
060, 
G76I 
686 


395 

o 385 
? 375 

o m 


35 

34 
33 
32 
31 
30 
29 
28 
27 
26 


25 

26 
27 
28 
29 
~30 
31 
32 
33 
34 


11 

11 

12 
12 
13 
13 

13 

n 

14 
15 


10 

11 

11 

12 
12 


7 

7 
8 
8 
8 


7 
7 

7 
7 
8 
~8~ 
8 
9 
9 
9 


6 

7 
7 

7 

~T" 

8 
8 
8 
8 


5 
5 
5 

5 
5 


4 
4 
4 
5 
5 


4 
4 
4 

4 
4 


c 09939 
, 5 914 
i 888 
* 862 
J 836 


89354 

S III 

I 
o 314 


12 
13 
13 
14 
14 


8 
9 
9 
9 
10 


6 


6 
6 


5 

5 
5 
6 
6 


4 
5 
5 
5 

5 


35 

36 
37 
38 
39 


494 
510 
526 
542 
558 


506 
490 
474 
458 
442 


190 2 

2ie; 
242; 

268 
294? 


o 81( 

a 784 
? 758 

A 732 

6 70 * 


(MX) 
706 
716 

726 
736 


o 304 
o 294 

% 

..i 274 

n aw 


, 
2^ 
2j 

21 
2 


35 

36 
37 
38 
39 


15 

10 
16 
10 
17 


15 
15 

15 
16 

10 


10 

10 
10 

11 
11 


9 
10 
10 
10 

10 


9 
9 

9 
10 
10 


(j 
7 
7 
7 
7 


6 
6 

6 
6 
6 


5 
5 
6 
6 
6 


40 

41 
42 
43 

44 


573 
589 
605 
621 
636 


427 
411 
395 
379 
364 
348 
332 
316 
301 
285 


320 
346, 
371 
397 
423^ 
449, 
475* 
601 
527 
553^ 


- 680 
J 654 
'I 629 
603 
577 


746, 
756 
767 
777 
787 


iH 

s 

n 213 


20 

19 


40 

41 
42 
43 
44 


17 
18 
18 
19 
19 
~20 
20 
20 
21 
21 


17 
17 

18 
18 
18 


11 

12 
12 

12 
12 


11 
11 

11 
11 

12 


10 

10 
10 

11 
11 


7 
8 
8 
8 
8 


7 
7 
7 

7 

7 


6 

6 
6 
6 

7 


45 

46 
47 
48 
49 


652 
668 
684 
699 
715 


f 551 
r 525 
A 499 
I 473 
I 447 


797. 
807 
817 
827 
838 J 


: s 

!? 3 

n 1B2 


45 

46 
47 

48 
49 


19 
19 
20 
20 
20 
"21 
21 
22 
22 
22 


13 
13 

13 
14 
14 
14 
14 
15 
15 
15 


12 

12 
13 
13 
13 
13" 
14 
14 
14 
14 


11 

12 
12 
12 

12 


8 
8 
9 
9 
9 


8 
8 
8 
8 

A 
~V 

8 
9 
9 
9 


! 

t 

7 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


731 
746 
762 
778 
793 


269 
254 
238 
222 
207 


578 
604' 
630 ; 
656" 
682 ^ 


. 422 
! 396 
? 370 

6 344 
318 


848, 
858 
868 
878 
888 


152 

o 142 
11 

!?i 


10 


50 

51 
52 
53 
54 


22 
22 
23 
23 
23 
24 
24 
25 
25 
26 
26 


12 
13 
13 
13 
14 
Tf 
14 
14 
14 
15 


9 
9 
10 
10 
10 


8 

8 
8 
8 

~fT 

8 
9 
9 
9 


809 
825 
840 
856 

872 


191 
175 
160 
144 
128 


708 r 
734 J 
759; 
785 ^ 
811 - 


292 
* 266 

r 5 241 
! 215 

6 189 


899, 
909 
919 
929 
940 


: s 
$ 

o 06 


55 

56 
57 

58 
59 


23 

23 
24 
24 
25 


10 
16 

16 
16 
17 


15 
15 

15 
15 
10 


10 

10 
10 

11 
11 


9 
9 
10 
10 
10 
10 


60 


79887 


20113 


90837' 


09163 


10950 


8905( 




60 


25 


17 


16 


15 


11 


9 


' 


9. 

1 COS 


10. 

I sec 


9. 

I cot 


d 10. 

l' Z tan 


10. 

/ CSC 


d 9. 

' I sin 






26 


25 


17 

Pro 


16 

portic 


15 

mal 


11 

Parts 


10 


9 


128 


51 

83 





39 C 



TABLE II 



140 C 



, nsm 
9. 


d I csc 
1' 10. 


I tan 
9. 1 


i I cot 
' 10, 


I sec 
10. 


d 
1 

10 




( 

10 
10 
10 







( 

o 

I 

11 

10 
10 
1 
10 

11 

10 
10 

11 

10 

11 

10 

11 

10 
10 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 

10 

11 
11 

10 

11 

10 

11 

10 

11 
11 


/ COS 

9. 


' 


// 


26 


Pro 
25 


portio 
16 


naJ P 
15 


irts 
11 


10 


079887 

1 903 
2 918 ] 
3 934 J 
4 960 


-30113 
* 097 
? 082 
! 066 
! 050 


90837 
863* 
889* 
914^ 
940* 


.09163 
! 137 

( ; in 

J 086 
060 


10950 
960 
970 
980 
991 


89050 
040 
030 
020 
009 


60 

59 
58 
57 
56 




I 

2 
3 

4 




1 

1 
2 




1 
1 
2 





1 
1 
1 






1 

1 






1 
1 







1 
~~Y 
i 

i 
i 

2 


5 965 

6 981 
7 996 
8 80012 
_9 027 


6 35 

? 019 

6 4 

! 19988 

5 Q73 
6 97d 


966 

992 1 
91018* 
043* 
069 g 


6 34 

! 008 
08982 

a 957 
J 931 


11001 
Oil 
022 
032 
042 
~052 
063 
073 
083 
094 


88999 
989 
978 
968 
958 
" 948 
937 
927 
917 
906 
89( 
886 
875 
865 
855 


55 

54 
53 
52 
51 


5 

6 

7 
8 
9 


2 

3 
3 
3 
4 


2 

2 
3 

3 

4 


1 

2 
2 
2 

2 


1 

2 
2 
2 
2 
2 
3 
3 
3 
4 


1 

1 
1 
1 

2 


1C 043 , 

11 058 
12 074 
13 089 
14 105 


. 957 

a 942 
: 926 

6 911 
5 895 


095 
121, 
147 1 
172, 
1 198^ 


, 905 
879 
I 853 
J 828 
J 802 


50 

49 
48 
47 
46 
45 
44 
43 
42 
41 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


4 
5 
5 
6 
6 


4 

5 
5 

5 
6 

6 
7 
7 

8 
8 


3 
3 

3 
3 

4 


2 
2 

2 
2 
3 


2 
2 
2 

2 
2 


15 120 , 

16 136 
17 151 
18 166 
19 182 j 


- 880 
, 864 

; 849 

6 834 

5 818 


224 
250, 
276, 
301* 
327 ?, 
353 
379, 
404^ 
430 : 
456* 


6 77C 

a 75 

5 724 

* 699 
J ti73 


104 
114 
125 
135 
145 


6 
7 

7 
8 
8 


4 

4 
5 
5 
5 


4 
4 
4 
4 
5 
5 
5 
6 
6 
6 


3 
3 

3 
3 
3 


2 
3 
3 
3 

3 


20 197 , 
21. 213 
22 228 1 
23 244 
24 259 } 


803 

J 787 

6 772 
5 756 

741 


6 647 

! 621 
ii 59H 

a 57 
544 


166 
166 
170 
187 
197 


844 
834 
824 
813 
803 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


9 
9 

10 

10 

10 


8 

g 
9 

10 

10 


5 
6 
6 

6 
6 


4 
4 
4 

4 
4 


3 
4 
4 
4 
_4 
4 
4 
4 
5 
5 


25 274 
26 290 J 
27 305 
28 320 J 
29 336 j 


6 72G 
5 71 

; 695 
680 
I 664 


482 
507, 
533, 
559* 

585 2 


518 

6 493 
467 

a 441 
? 415 


207 
218 
228 
239 
249 


793 

782 
772 
761 
751 


35 

34 
33 
32 
31 


11 

11 

12 
12 

13 


10 

11 

11 

12 
12 


7 
7 

7 
7 
8 


6 
6 
7 
7 

7 


5 
5 
5 

5 
5 


3080351, 
31 366 
32 382 
33 397 ) 
34 412 


, 19649 
* 634 

5 618 

f 603 

ij 588 


91610 , 
636, 
662, 
688, 
713* 


,. 08390 

6 3M 

: 338 

* 312 
s 287 


11259 
270 
280 
291 
301 


88741 
730 
720 
709 
699 


30 

29 
28 
27 
26 
25 
24 
23 
22 
21 


30 

31 
32 
33 
34 


13 

13 
14 

14 
15 


12 
13 

13 
14 
14 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


6 
6 
6 
6 
6 


5 

5 
5 
6 
6 


35 428, 
36 443 
37 458 } 
38 473 
39 489 } 


, 572 
! 557 
I 542 

6 627 
J 511 


739 9 
765; 
791, 
816* 

842;; 


8 261 

! 235 
. 209 

B 184 
158 


312 
322 
332 
343 
353 


688 
678 

ens 

657 
647 


35 

36 
37 
38 
39 
40 
41 
42 
43 
44 


15 

16 
16 
16 
17 


15 
15 

15 
16 

16 



10 
10 
10 
10 


9 
9 

9 
10 
10 


6 
7 

7 
7 
7 


6 
6 

6 
6 
6 


40 504 , 
41 519 1 
42 534 
43 550 
44 565 J 


, 496 
? 481 
f 466 
I 450 
f 435 


868 
893* 
919 1 

945 1 
971^ 


5 132 

107 

R 081 
, 055 
, 029 


3b'4 
374 
385 
395 
406 


63(5 
026 
615 
605 
594 


20 

10 
18 
17 
16 


17 
18 
18 
19 
19 


17 
17 

18 
18 

18 


11 
11 

11 
11 

12 


10 

10 
10 

11 
11 


7 
8 
8 
8 
8 


7 
7 
7 

7 
7 


45 580 , 
46 595 } 
47 610 
48 625 
49 641 


5 42 

I 405 
* 390 
37.5 

5 359 


996 
92022* 
048* 
073* 
099* 


B 4 

07978 
J 952 

5 927 

b 901 
"87J5 
850 
J 824 
798 
J 773 


416 
427 
437 

448 
458 


584 
573 
563 
552 
542 


15 

14 
13 
12 
11 
10 
9 
8 
i 
6 

5 

4 

r 
i 

1 


45 

4(i 
47 
48 
49 


20 
20 
20 
21 

21 


19 
19 
20 
20 

20 


12 

12 
13 
13 
13 


11 

12 
12 
12 

12 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


5( 656 , 
51 671 
52 686 
53 701 J 
54 716} 


, 344 
? 329 
J 314 

: 299 

J 284 


125 
150 1 
176 1 
202* 
227 1 


469 
479 
490 
501 
511 


531 
521 
510 
499 
489 


50 

51 
52 
53 

54 


22 
22 
23 
23 

23 


21 

21 
22 
22 

22 


13 
14 
14 

14 
14 


12 
13 
13 

13 
14 
14 
14 
14 
14 
15 


9 
9 
10 
10 
10 


8 
8 
9 
9 
9 


55 731 , 

56 746 
57 762 J 
58 777 
59 792 } 


, 269 
J 254 
! 238 
J 223 
: 208 


253 _ 
279 * 
304 1 
330 J 
366? 


3 747 

: 721 

: 696 

' 670 
? 644 


522 
532 
543 
553 
564 


478 
468 
457 
447 
436 


55 

56 
57 

58 
59 


24 

24 
25 
25 

26 


23 

23 
24 
24 

25 


15 
15 

15 
15 
16 


10 

10 
10 

11 
11 


9 
9 
10 
10 
10 


Ofl 80807 


5 19193 


92381 


07619 


11575 


88425 
9." 

I sin 




f 


60 


26 


25 


16 


15 


11 
11 

its 


10 
10 


, 9. d 

I COS 1 


10. 

' Zscc 


9. d 

I cot i 


10. 

Z tan 


10. 

1 CSC 


d 
r 


n 


26 


25 

Pro 


16 

portioi 


15 

lalPa 



129 



50 

84 



40 C 



TABLE II 



139 C 



, / sill d 


I CSC 

10. 


tan d 
9. 1' 


I cot 
10. 


sec d 
10. 1 


1 COS , 

9. 


n 


Proi 
26 25 


lortioi 
15 


tal Pa 
14 


rts 
11 


10 


80807 
1 822 1; 
2 837 }' 
3 852 ' 
4 867 !' 


19193 
178 
' 163 
J 148 
? 133 


2381 _ 
407?! 
433^ 
458 o 
484 1 


07619 
593 
567 
542 
516 


1575 
585 1? 
596 
606 

617 


88425 60 
41559 
40458 
39457 
383 Ffi 




1 

2 
3 

4 




1 
1 
2 




1 

1 
2 






1 
1 






1 
1 






1 
1 






1 


5 882, 
6 897; 
7 912, 
8 927 
9 942 


\ 118 

; 103 

? 088 
J 073 
1 058 
' 043 
028 
013 
i 18998 

5 983 


510 ; r 
535 1 

561 ; 

587^ 
612 J 


490 
465 
439 
413 
388 
362 
337 
311 
285 
260 


628 

US" 

660 ; ( 

670 < 


, 372 55 

} 362 54 
35153 
, 340 52 
} 33051 


5 

6 
7 

8 
9 


2 

3 
3 
3 
4 


2 

2 
3 

3 
4 


1 

2 
2 
2 

2 


1 
1 

2 
2 
2 


1 

1 
1 

1 

2 


1 
1 

1 
1 

2 


10 957 , 

11 972 
12 987 
13 81002 j 
14 017) 


638 ' 
663 2 ,; 
689 1 
715 ? 

740 ;; 


681 
692 
702 ( 
713 
724 


31950 
308 49 
} 298 48 
1 287 47 
\ 276 46 


10 

11 
12 
13 
14 


4 
5 
5 
6 
6 


4 

5 
5 
5 
6 


2 
3 
3 

3 

4 


2 
3 
3 
3 
3 


2 
2 

2 
2 
3 


2 
2 
2 

2 
2 


15 032 

16 047 
17 061 
18 076 } 
19 091 1 


R 968 

A 953 
t 939 

5 924 
J 909 


766. 
792 2 2 
817* 
843;5 
868; 


234 
208 
183 
157 

_J 32 
106 
080 
055 
029 
004 


734 
745 
756 
766 

777! 


26645 
255 44 
24443 
\ 234 42 
223 41 


15 

16 
17 
18 
19 


6 
7 

7 
8 
8 


6 

7 
7 
8 
8 


4 
4 

4 
4 
5 
5 
5 
6 
6 
6 


4 
4 
4 

4 
4 


3 
3 

3 
3 
3 


2 
3 
3 
3 

3 


250 106 , 
21 121 
22 136 
23 151 
24 166 


r 894 
^ 879 
? 864 
J 849 
4 834 


894, 
920^ 
945^ 
971^ 
996; 


788 , 
799 
809 J 
S20 
831} 


21240 

\ 20139 
? 191 38 
18037 
J 16936 


20 

21 
22 
23 

24 


9 
9 
10 
10 

10 


8 
9 
9 
10 
10 


5 
5 

5 
5 
6 


4 
4 
4 

4 
4 


3 
4 
4 
4 
4 


25 180, 
26 195 
27 210; 
28 225 
29 240 ; 
30 81254 , 
31 269 
32 284 
33 299 
34 314 


r 820 
; 805 
J 790 
' 775 
* 760 


93022 
048 2 

073; 

099, 
124 2 


06978 
952 
927 
J 901 

1 87( 


842 
852 1 
863 | 
874 
885! j 
11895L 
906| 
917' 
928 
939 J 


, 15835 
148 34 
1 13733 
12632 

i.n?3 

,881053 
I 0942 
0832 
0722 
0612 


25 

26 
27 

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29 


11 

11 

12 
12 

13 


10 

11 

11 

12 
12 


6 
?> 
7 
7 

7 


6 
6 

6 

7 
7 


5 
5 
5 
5 
5 


4 
4 
4 
5 
5 


1874 
J 73 

; J 71 

'1 70 

4 6 8 


93J50, 
175 \ 

201 a 

227^ 

252; 


00850 

o 82 
79 

'.'i 77 
74 


30 

31 
32 
33 
34 


13 

13 
14 

14 
15 


12 
13 
13 
14 
14 


8 
8 
8 
8 
8 


7 

7 
7 
8 
8 


6 
6 
6 
6 

6 


5 

5 
5 
6 
6 


36 328 , 
36 343 ' 
37 358 
38 372 ] 
39 387 


c 67 
? 65 

4 <> 4 

* 62 
5 61 


278" 
303^ 
329~ 
354" 

380;; 


r 72 

f 69 
'' 67 
; 64 
62 


949, 
960| 
971 ] 
982] ] 
993i 


. 0512 
| 0402 
i 0292 
0182 
J! 0072 
, ',87996 -r 
- ^R^ 
\ 975 
964 
953 


35 

36 
37 

38 
39 


15 

10 
16 
lb 
17 


15 
15 

15 
16 

10 
17 
17 

18 
18 

18 


9 
9 

9 
10 
10 


8 
8 
9 
9 
9 


6 
7 
7 
7 

7 


6 
6 

6 
6 
6 


40 402 

41 417 
42 431 
43 446 

44 461 


6 5 
4 ^ 

t 56 
1 55 

: 53 


406 
431, 
457 2 

482" 
508* 


59 
* 56 
54 

c 61 
1 49 


12004 
015 
025 
036 
047 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


17 
18 
18 
19 
19 

' 20 

20 

20 
21 

21 


10 

10 
10 

11 
11 

11 

12 
12 
12 

12 


9 
10 
10 
10 

10__ 
10 

11 
11 

11 
11 


7 
8 
8 
8 
8 


7 
7 
7 

7 

7 


45 - 475 
46 490 
47 505 
48 519 
49 534 


52 
5 5l 

14 49 

L J 48 
l " 46 


533; 
559? 
584? 
610? 
636? 
661 
687? 
712? 
738; 
763? 


:s 

1 39 
36 


058 
069 
080 
091 
102 


L 42 

L 931 
920 

s 


19 
19 
20 
20 

20 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


50 549 

51 563 
52 578 
53 592 
54 607 


14 45 
1 J 43 

4 42 

1 J 40 

5 39 
4~37 
l ? 36 
1 J 34 
J 33 
1 J 32 


! 3 

d 28 
>6 26 
25 23 

>6 


113 
123 
134 
145 
156 

""167 
178 
189 
200 
211 


^ gg' 

i S 

844 


50 

51 
52 
53 
54 


22 
22 
23 
23 

23 


21 

21 
22 
22 

22 


12 
13 
13 

13 

14 


12 
12 

12 
12 
13 


9 
9 
10 
10 
10 


8 
8 
9 
9 
9 


55 622 

56 636 
57 651 
58 665 
59 680 


789; 
814; 
840 ; 
865; 

891 ; 

W9l6' 


s 

s is 

>r *" 
5 l 

0008- 


833 
1 822 

811 
800 
789 

87778 


55 

56 
57 
58 
59 


24 

24 
25 
25 

2(3 


23 

23 
24 
24 

25 


14 
14 

14 
14 
15 


13 
13 

13 
14 
14 


10 

10 
10 

11 
11 


9 
9 
10 
10 
10 


00 81694 


1830 


12222 


60 


26 


25 


15 


14 


11 


10 


, 9. 

I cos 


d 10. 

1' 1 KT 


9. 

I cot 


d 10. 

i' Ztfti 


10. 

/ osc 


d 9. 

i' / sin 


n 


26 


25 

Pr 


15 

oportif 


14 

nalP 


11 

arts. 


10 


130 49 

85 



41 C 



TABLE II 



138 



' 


I sin 
9. 


d 
i 


1 CSC 

10. 


Uan 
9. 


d 


I cot 
10. 


I sec 
10. 


d 

i' 


1 COS 

9. 


f 


// 


26 


1*0 

25 


portio 
15 


nalPa 
14 


rts 
12 


11 




1 

2 
3 

4 
5 

6 
7 
8 
9 

To 

11 

e 

Is 

14 


81694 
709 
723 
738 
752 


5 
14 
15 
14 
15 
14 
15 
14 
15 
4 


18306 
291 
277 
262 

248 


93916 
942 
967 
993 
94018 


26 
25 
26 
25 
26 
25 
26 

26 
25 
26 

26 
25 
26 

26 
23 
26 

26 
25 
26 
25 
26 
25 
25 
26 
25 
26 

26 
25 
26 
25 
26 
25 
25 
26 
25 
26 
25 
26 
25 
25 
26 
25 
26 
25 
26 
25 
25 
26 
25 
26 
25 
26 
25 
25 
26 


06084 
058 
033 
007 
05982 


12222 
233 
244 
255 
266 


n 
n 
n 
n 
n 
n 
11 
11 
11 
11 
n 
n 
n 
n 
n 

12 
11 
11 
11 
11 
11 
11 
11 
11 
12 
11 
11 
1] 
11 
11 

u 
11 
11 
11 

11 
11 

1 ] 


87778 
767 
756 
745 
734 


60 

59 

58 
57 
56 




1 

2 
3 

4 




1 
1 
2 





1 
1 
2 






1 

1 







1 

1 






1 
1 






1 
1 


767 
781 
796 
810 
825 


233 
219 
204 
190 
175 


044 
069 
095 
120 
146 
171" 
197 
222 
248 
273 


956 
931 
905 
880 

854 


277 
288 
299 
310 
321 


723 
712 
701 
690 
679 


55 

54 

K 

52 
51 


5 

6 

7 
8 
9 


2 

3 
3 
3 
4 


2 

2 
3 

3 
4 


1 

2 
o 

2 

2 


1 
1 

2 
2 
2 
2 
3 
3 
3 
3 


1 

1 
1 

2 
2 


1 

1 
1 
1 

2 


839 
854 
868 
882 
897 


15 
14 
14 
15 


161 
146 
132 
118 
103 


829 
803 
778 
752 

727 


332 
343 
354 
365 
376 


668 
657 
646 
635 
624 


50 

49 
48 
47 
46 


10 

11 
12 
13 

14 


4 
5 
5 
6 
6 


4 
5 
5 

5 
6 


2 
3 
3 

3 

4 


2 

2 
2 
3 
3 


2 
2 

2 
2 
3 


15 

L ( 
.< 
18 
19 
20 
21 
22 
23 
24 


911 
926 
940 
955 
969 


14 
14 


089 
074 
060 
045 
031 


299 
324 
350 
375 
401 


701 
676 
650 
625 
599 
574 
548 
523 
497 
472 


387 
399 
410 
421 
432 
443 
454 
465 
476 
487 


613 
601 
590 
579 
568 


45 

44 
43 
42 
41 


15 

16 
17 
18 
19 


6 
7 

7 
8 
8 


6 
7 
7 

8 
8 


4 
4 

4 
4 
5 


4 
4 
4 
4 

4 


3 

3 
3 
4 

4 


3 
3 

3 
3 
3 


983 
998 
82012 
026 
041 


15 

14 
15 
14 
[4 
15 
14 
14 
11 


017 
002 

17988 
974 
959 


426 
452 
477 
503 
528 


557 
546 
535 
524 
513 


40 

39 
38 
37 
36 


20 

21 
22 
23 
24 


9 
9 
10 
10 

10 


8 
9 
9 
10 
10 


5 

5 
6 
6 
6 


5 
5 

5 
5 
6 


4 

4 
4 
5 
5 


4 
4 
4 

4 
4 


25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


055 
069 
084 
098 
112 


945 
931 
916 
902 

888 


554 
579 
604 
630 
655 


446 
421 
396 
370 
345 


499 
510 
521 
532 
543 


501 
490 
479 
468 
457 
87446 
434 
423 
412 
401 


35 

34 
33 
32 
31 


25 

26 
27 
28 
29 


11 

11 

12 
12 

13 


10 

11 

n 

12 
12 


6 
6 
7 
7 

7 


6 
6 

6 
7 

7 


5 

5 
5 
6 
6 


5 
5 
5 
5 
5 


82126 
141 
155 
169 
184 


15 
14 
14 
15 
11 


17874 
859 
845 
831 
81f 


94681 
706 
732 
757 

783 


05319 
294 
268 
243 

217 


12554 
566 

577 
588 
599 


30 

29 
28 
27 
26 


30 

31 
32 
33 
34 


13 

13 
14 

14 
15 


12 
13 
13 
14 
14 


8 
8 
8 
8 
8 


7 

7 
7 
8 
8 


6 

G 
6 

7 
7 


6 
6 
6 
6 

6 


35 

36 
37 
38 
39 


198 
212 
226 
240 
255 


14 
14 
14 
15 
14 

14 
14 
15 
14 
14 
14 
14 
14 
14 
14 
15 
14 
14 
11 


802 
788 
774 
760 
745 


808 
834 
859 
884 
910 


192 
166 
141 
llf 
090 


610 
622 
633 
644 
655 


390 
378 
367 
35f 
345 


25 

24 
23 
22 
21 


35 

30 
37 
38 
39 


15 

16 
16 
16 
17 


15 
15 
15 
16 
16 
17 
17 
18 
18 
18 
19 
19 
20 
20 
20 


g 
9 

9 
10 
10 


8 
8 
9 
9 
9 


7 

7 
7 
8 
8 


6 
7 
7 
7 

7 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


269 
283 
297 
311 
326 


731 
717 
703 
689 
674 


935 
961 
98f 
95012 
037 


065 
039 
014 
04988 
963 


660 
678 
689 
700 
712 


12 
11 
11 
12 
11 
11 
11 
12 
11 
11 
12 

1! 
11 

1 


334 
322 
31 
300 
288 
277 
266 
255 
243 
232 


20 

19 
18 
17 
16 


40 

41 
42 
43 
44 


17 
18 
18 
19 
19 
20 
20 
20 
21 
21 
"22"~ 
22 
23 
23 
23 
~2F 
24 
25 
25 
26 


10 

10 
10 

11 
11 

11 

12 
12 
12 

12 


9 
10 
10 
10 

10 


8 
8 
8 
9 
9 


7 
8 
8 
8 
8 


340 
354 
368 
382 
396 


660 
646 
632 
618 
604 


062 
088 
113 
139 
164 


938 
912 
887 
861 
836 


723 
734 
745 
757 

768 


15 

14 
13 
12 
11 


45 

46 
47 
48 
49 


10 

11 
11 

11 
11 


9 

9 
9 
10 
10 


8 
8 
9 
9 
9 


50 

51 
52 
53 
54 


410 
424 
439 
453 
467 


590 
576 
561 
547 
533 


190 
215 
240 
266 
291 


810 
785 
760 
734 
709 


779 
791 
802 
813 
825 


22 
20<3 
198 
187 
175 


10 


50 

51 
52 
53 
54 


21 

21 
22 
22 

22 


12 
13 
13 

13 
14 


12 
12 

12 
12 
13 


10 

10 
10 

11 
11 


9 
9 
10 
10 
10 


55 

56 
57 

58 
59 


481 
495 
509 
523 
537 


14 
14 
14 
14 
14 


519 
505 
491 
477 
463 


317 
342 
368 
393 

418 


683 
658 
632 
607 
582 


836 
847 
859 
870 
881 


164 
153 
14 
13C 
11C 




55 

56 
57 

58 
59 


23 

23 
24 
24 

25 


14 
14 

14 
14 
15 


13 
13 

13 
14 
14 


11 

11 
11 

12 
12 


10 

10 
10 

11 
11 


60 


82551 


17449 


9544' 


0455G 


12893 




87107 





60 


26 


25 


15 


14 


12 


11 


/ 


2 cos 


d 
1 


10. 

I sec 


9. 

I cot 


c 
1 


10. 

I tan 


10. 

I CSC 


d 
1 


9. 

2 sin 


' 




26 


25 

Pr< 


15 

>porti( 


14 

nalP 


12 

aits 


11 


131 


48 

86 





42 C 



TABLE II 



137 C 





1 

2 
3 

4 


I sin 
9. 


d 

1' 


I CSC 

10. 


Ztan 
9. 


d 


Zcot 
10. 


I sec 
10. 


d 


/ COS 

9. 


/ 

60 

59 
K 
)7 
56 
55 
K 

K 

52 
51 




/, 


96 


Pro 
95 


>ortioi 
14 


lalPa 
13 


rts 
19 


11 


89551 
565 
579 
593 
607 


14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
13 
14 
14 
14 
14 
14 
14 
13 
14 
14 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
IS 
14 
14 
13 
14 
14 
13 
14 
13 


17449 
435 
421 
407 
393 


15444 


25 
26 
25 
25 
26 
25 
26 
25 

26 
25 
25 
26 
25 
26 
25 
25 
26 

26 
25 
25 
26 

25 
2G 

26 
25 

25 
26 

25 
26 
25 
25 
26 
25 
21 
26 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 

c 

1 


04556 
531 
505 
480 
455 


[9893 


1 

1 

2 
11 
9 


87107 
096 
085 
073 
062 






1 

2 
3 

4 




1 

1 

2 




1 

1 
2 






1 
1 






1 
1 






1 
1 






1 
1 


469 
495 
520 
545 


904 
915 
927 
938 


6 

6 
7 
8 
9 


621 
635 
649 
663 
677 


379 
365 
351 
337 
323 


571 
596 
622 
647 
672 


429 
404 
378 
353 
328 


950 
961 
972 
984 
995 


11 
11 
12 
11 
12 
11 
12 
11 
12 
11 
12 
11 
11 
12 
11 

12 
12 
11 
12 
11 

11 
12 
11 
12 
11 
12 
12 
11 
12 
11 
12 
12 
11 
12 

11 
12 
12 
11 


050 
039 
028 
016 
005 


5 

6 

7 
8 
9 


9 

3 
3 
3 
4 


9 

2 
3 

3 
4 


1 
1 

2 
2 
9 


1 

1 

2 
2 
9 


1 

1 
1 

2 
2 


1 

1 
1 
1 

2 


10 

1 
2 

,3 
14 


691 
705 
719 
733 
747 


309 
295 
281 
267 
253 


698 
723 

748 
774 
799 


302 
277 
252 
226 
201 


13007 
018 
030 
041 
053 


86993 
982 
970 
959 
947 


50 

49 

48 
47 
46 


10 

11 
12 
13 
14 


4 
5 
5 
6 
6 


4 

5 
5 

5 
6 


2 
3 
3 
3' 

3 


2 
2 
3 
3 
3 


9 

2 
2 
3 
3 


2 
9 

2 
2 
3 


15 

6 
7 
18 
19 
90 
11 
!2 
23 
24 
95 
26 
27 
28 
29 


761 
775 
788 
802 
816 


239 
225 
212 
198 
184 
"170 
156 
142 
128 
115 
101 
087 
073 
059 
045 


825 
850 
875 
901 
926 


175 
150 
125 
099 
074 


064 
076 
087 
098 
110 


936 
924 
913 
902 
890 
~879 
867 
855 
844 
832 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 


15 

16 
17 
18 
19 


6 
7 

7 
8 
8 


6 
7 
7 

8 
8 


4 
4 
4 

4 
4 


3 
3 
4 
4 

4 


3 

3 
3 
4 
4 


3 
3 

3 
3 
3 


830 

844 
858 
872 
885 


952 
977 
96002 
028 
053 
078 
104 
129 
155 
180 


048 
023 
03998 
972 
947 
922 
896 
871 
845 
820 


121 
133 
145 
156 
168 
"179 
191 
202 
214 
225 


90 

21 
22 
23 
24 


9 
9 
10 
10 

10 


8 
9 
9 
10 
10 


5 
5 

5 
5 

6 


4 
5 
5 
5 
5 


4 

4 
4 
5 
5 


4 
4 
4 

4 
4 


899 
913 
927 
941 
955 


821 
809 
798 

786 

775 


95 

26 
27 
28 
29 


11 

11 

12 
19 

13 


10 

11 

11 

12 
19 


6 
6 

6 

7 
7 


5 
6 
6 
6 
6 


5 

5 
5 

6 
6 


5 
5 
5 

5 
5 


30 

31 

V 

35 
34 


89968 
982 
996 
83010 
023 


17032 
018 
004 
16990 
977 


96205 
231 
256 
281 
307 


03795 
769 
744 
719 
693 


13237 
248 
260 
272 
283 


86763 
752 

740 
728 
717 


30 

29 
28 
27 
26 
95 
24 
23 
22 
21 


30 

31 
32 
33 
34 


13 

13 
14 

14 
15 


12 
13 
13 
14 
14 


7 

7 
7 
8 
8 


6 

7 
7 

7 
7 


6 

6 
6 
7 
7 


6 
6 
6 
6 
6 


35 

36 
37 
38 
39 


037 
051 
065 
078 
092 


963 
949 
935 
922 
908 


332 
357 
383 
408 
433 


668 
643 
617 
592 
567 


295 
306 
318 
330 
341 


705 
694 
682 
670 
659 


35 

3(> 
37 
38 
39 


15 

16 
16 
16 
17 


15 
15 

15 
16 

16 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


7 

7 
7 
8 
8 


6 
7 
7 
7 

7 


40 

41 
42 
43 
44 


106 
120 
133 
147 
161 


894 
880 
867 
853 
839 


459 
484 
510 
535 
560 


541 
516 
490 
465 
440 


353 
365 
376 
388 
400 


647 
635 
624 
612 
600 


90 

19 
18 
17 
16 
15 
14 
13 
12 
1 


40 

41 
42 
43 
44 


17 
18 
18 
19 
19 


17 
17 

18 
18 
18 


9 
10 
10 
10 
10 


9 
9 
9 
9 
10 


8 
8 
8 
9 
9 


7 
8 
8 
8 
8 


45 

46 
47 
48 
49 
50 
51 
52 
53 
54 


174 
188 
202 
215 
229 


826 
812 
798 
785 
771 


586 
611 
636 
662 
687 


414 
389 
364 
338 
313 


411 
423 
435 
44C 
458 


12 
12 
11 
12 
12 
12 
11 

12 

12 
12 
12 
1 
12 

d 
1 


589 
577 
565 
554 
542 


45 

46 
47 

48 
49 


20 
90 
20 
91 

21 


19 
19 
20 
90 

20 


10 

11 
11 

11 
11 


10 
10 

10 
10 

11 


9 

9 
9 
10 
10 


8 
8 
9 
9 
9 


242 
256 
270 
283 
297 


14 
14 
13 

13 
14 
14 
13 
14 
13 

7 
1 


758 
744 
730 
717 
703 


712 
738 
763 
788 
814 


288 
262 
237 
212 

186 


470 
482 
493 
505 

517 


530 
518 
507 

495 

483 


10 


50 

51 
52 
53 
54 


22 
99 
23 
93 

23 


91 

21 
22 
99 

22 


12 
19 

12 
12 
13 


11 
11 

11 
11 

12 


10 

10 
10 

11 
11 


9 
9 
10 
10 
10 


55 

56 
57 
58 
59 


310 
324 
338 
351 
365 


690 
676 
662 
649 
635 


839 
864 
890 
915 
940 


1-6 
136 
110 
085 
060 


528 
540 
552 
564 
575 


472 
46C 
448 
436 
t425 




55 

56 
57 

58 
59 


94 

24 
25 
95 

26 


93 

23 
24 
94 

25 


13 
13 

13 
14 
14 


19 

12 
12 
13 
13 


11 

11 
11 

12 
12 


10 

10 
10 

11 
11 


60 


83378 


1662? 


9696f 


03034 


13587 


86413 




60 


96 


95 


14 


13 


19 


11 


' 


9. 

1 COS 


10. 

Zsec 


9. 

Zcot 


10. 

Han 


10. 

I CSC 


9. 

/ sin 


' 


n 


96 


95 

Pr 


14 

>portio 


13 

nalP 


1 

urts 


11 


132 


47 

87 



43 C 



TABLE II 



136 C 



/ 


Ism 
9. 


d 
1' 

14 
13 
14 
13 
14 
13 
14 
13 
14 
13 
14 
13 
14 
13 
14 


Zcsc 
10. 


I tan 
9. 


d 
1' 


I cot 
10. 


I sec 
10. 


d 
1' 


1 COS 

9. 


/ 




n 


26 


Pro 
25 


po ? o 


aal Pa 
13 


rts 
12 


11 




1 

2 
3 
4 


83378 
392 
405 
419 
432 


16622 
608 
595 
581 

568 


96966 
991 
97016 
042 
067 


25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 

25 
25 

26 
25 
25 
26 
25 
23 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 
25 
25 
26 
25 
25 
25 
26 


03034 
009 
02984 
958 
933 


13587 
599 
611 
623 
634 


12 
12 
12 
11 
12 
12 
12 
12 
12 
11 
12 
12 
12 
12 
12 
12 
12 
11 
12 
12 
12 
12 
12 
12 

12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 
13 
12 
12 
12 
12 
12 
12 
13 
12 
12 
12 
12 
12 
13 


86413 
401 
389 
377 
366 


60 

59 
58 
57 

56 






1 

2 
3 

4 




1 
1 
2 





1 
1 
2 






1 
1 






1 
1 






1 
1 






1 
1 


5 

6 

7 
8 
9 


446 
459 
473 
486 
500 


554 
541 
527 
514 
500 


092 
' 118 
143 
168 
193 


908 
882 
857 
832 
807 


646 
658 
670 
682 
694 


354 
342 
330 
318 
306 


55 

54 
53 
52 

51 


5 

6 

7 
8 
9 


2 

3 
3 
3 
4 


2 

2 
3 

3 
4 


1 
1 

2 
2 
2 


1 

1 

2 
2 
2 


1 
1 

1 

2 
2 


1 
1 
1 

1 

2 


10 

11 
12 
13 
14 


513 
527 
540 
554 
567 


487 
473 
460 
446 
433 


219 
244 
269 
295 
320 


781 
756 
731 
705 
680 

~~655 
629 
604 
579 
553 


705 
717 
729 
741 
753 


295 
283 
271 
259 

247 


50 

49 

48 
47 
46 


10 

11 
12 
13 
14 


4 
5 
5 
6 
6 


4 
5 
5 
5 
6 


2 
3 
3 
3 

3 


2 
2 
3 
3 
3 


2 

2 
2 
3 
3 


2 
2 

2 
2 
3 


15 

16 
17 
18 
19 
20 
21 
22 
23 
24 


581 
594 
608 
621 
634 


13 
14 
13 
13 
14 
13 
13 
14 
13 
14 
13 
13 
14 
13 
13 
14 
13 
13 
13 
14 
13 
13 
13 
14 
13 
13 
13 
14 
13 
13 
13 
13 
14 
13 
13 
13 
13 
13 
13 
14 
13 
13 
13 
13 
13 


419 
406 
392 
379 
366 
352 
339 
326 
312 
299 


345 

371 
396 
421 
447 


765 
777 
789 
800 
812 


235 
223 
211 
200 

188 


45 

44 
43 
42 
41 
40 
39 
38 
37 
36 




15 

16 
17 
18 
19 


6 
7 

7 
8 
8 


6 
7 
7 

8 
8 


4 
4 
4 
4 
4 


3 
3 
4 
4 
4 


3 

3 
3 

4 
4 


3 
3 

3 
3 
3 


648 
661 
674 
688 
701 


472 
497 
523 
548 
573 


528 
503 
477 
452 
427 


824 
836 
848 
8(iO 
872 


176 
164 
152 
140 
128 


20 

21 
22 
23 
24 


9 
9 
10 
10 

10 


8 
9 
9 
10 
10 


5 
5 

5 
5 

6 


4 
5 
5 
5 
5 


4 

4 
4 
5 
5 


4 
4 
4 

4 
4 


25 

26 
27 
28 
29 


715 
728 
741 
755 

768 


285 
272 
259 
245 
232 


598 
624 
649 
674 
700 


402 
37H 
351 
326 
300 


884 
896 
908 
920 
932 


116 
104 
092 
080 
068 
86056 
044 
032 
020 
008 
85996 
984 
972 
960 
948 


35 

34 
33 
32 
31 

3d 

29 
28 
27 
2G 


25 

26 
27 
28 
29 


11 

11 

12 
12 

13 


10 

11 

11 

12 
12 


6 
6 

6 

7 

7 


5 
6 
6 
6 

G 


5 

5 
5 
6 
G 


5 
5 
5 

5 
5 


30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 


83781 
795 
808 
821 
834 


16219 
205 
192 
179 
166 


97725 
750 
776 
801 
826 


02275 
250 
224 

199 
174 


13944 
956 
908 
980 
992 


30 

31 
32 
33 
34 


13 

13 
14 

14 
15 


12 
13 

13 
14 
14 


7 

7 
7 
8 
8 


6 
7 
7 

7 

7 


6 

G 
6 

7 
7 


6 
G 
6 
6 
6 


848 
861 
874 
887 
901 


152 
139 
126 
113 
099 
086 
073 
060 
046 
033 


851 
877 
902 
927 
953 
978 
98003 
029 
054 
079 


149 
123 
098 
073 
047 


14004 
016 
028 
040 
052 
"064 
076 
088 
100 
112 


25 

24 
23 
22 

21 

20 

U 

18 
17 
16 


35 

36 
37 
38 
39 


15 

1C 
16 
16 
17 


15 
15 
15 
16 

16 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


7 

7 
7 
8 
8 


6 
7 
7 
7 

7 
7 
8 
8 
8 
8 


914 
927 
940 
954 
967 


022 
01997 
971 
946 
921 


936 
924 
912 
900 

888 


40 

41 
42 
43 
44 


17 
18 
18 
19 
19 


17 
17 

18 
IS 

18 


9 
10 
10 
10 

10 


9 
9 
9 

9 
10 


8 
8 
8 
9 
9 


45 

46 
47 
48 
49 


980 
993 
84006 
020 
033 


020 
007 
15994 
980 
967 


104 
130 
155 
180 
206 


896 
870 
845 
820 
794 


124 
136 
149 
161 
173 


876 
864 
851 
839 
827 


15 

14 
13 
12 
11 


45 

46 

47 
48 
49 


20 
20 
20 
21 

21 


19 
19 
20 
20 

20 


10 

11 
11 

11 
11 


10 
10 

10 
10 

11 


9 

9 
9 
10 
10 
10 
10 
10 

11 
11 


8 
8 
9 
9 
9 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


046 
059 
072 
085 
098 


954 
941 
928 
915 
902 


231 
256 
281 
307 
332 


769 
744 
719 
693 
668 


185 
197 
209 
221 
234 


815 
803 
791 
779 
766 
754 
742 
730 
718 
706 


10 

9 

8 

6 


50 

51 
52 
53 
54 


22 
22 
23 
23 

23 


21 

21 
22 
22 

22 


12 
12 

12 
12 
13 


11 
11 

11 
11 

12 


9 
9 
10 
10 
10 


112 
125 
138 
151 
164 


888 
875 
862 
849 
836 


357 
383 
408 
433 

458 


643 
617 
592 
567 
542 


246 
258 
270 
282 
294 


( 


55 

56 
57 
58 
59 


24 

24 
25 
25 

26 


.23 

23 
24 
24 

25 


13 
13 

13 
14 
14 


12 

12 
12 
13 
13 


11 

11 
11 

12 

12 


10 

10 
10 

11 
11 


60 

/ 


84177 


15823 


98484 


01516 


14307 


85693 





60 


26 


25 


14 


13 


12 


11 


9. 

1 COS 


d 

1' 


10. 

Zsec 


9. 

Zcot 


d 

1' 


10. 

Ztan 


10. 

I CSC 


d 
1 


9. 

I sin 


t 




26 


25 

Pr 


14 

oporti 


13 

onalP 


12 

arts 


11 


133 46 

88 



44 



TABLE II 



135 C 





I sin d 
9. i 


I CSC 

' 10. 


tan d 
9. i 


I cot 
' 10. 


J 80C d 

10 i 


Tcos f 
' 9. 


it 


26 


Propo 
25 


rtional 
14 


Parts 
13 


12 




1 

2 
3 

4 


84177. 
190 
203? 
216 
229 


, 15823 
810 
797 
I 784 
771 


98484 Of 
509 2 ; 

634; 

560 * 
686 J 


01516 
1 491 
! 466 
1 440 
! 415 


4307 ir 

319 ; 

331 ; 
343}: 
356?' 


,8569360 
681 59 
669 58 
657 57 
64556 




1 
2 
3 

4 




1 
1 
2 





1 
1 
2 






1 

1 






1 
1 






1 
1 


5 

6 
7 

8 
9 


242 . 
255 
269 
282 
295 


758 
* 745 

; 731 

J 718 
705 


610" 
6352. 
661 2< 
686 
7112. 


390 
'> 365 
' 339 
> 314 

5 289 


368 / 
380 1' 

392 ; 

404 J 

417 


, 632 55 

620 54 
608 53 
59652 
; 583 51 


5 

6 
7 
8 
9 


2 

3 
3 
3 
4 


2 

2 
3 

3 
4 


1 

1 

2 
2 
2 


1 

1 

2 
2 
2 


1 

1 
1 

2 
2 


10 

1 

2 
g 

!<4 
15 

C 
.7 
18 
c 

21 

21 
22 
23 
24 


308, 
321 
334 
347; 
360 J 


, 692 
* 679 
I 666 
! 653 
? 640 
o 627 
2 615 
f 602 
I 589 
I 576 


737 2i 
7622 
7872 
8122 
8382 


1 263 
5 238 
5 213 
5 188 
5 162 


429 , 
441 
453 
466 
478 


: 571 50 

55949 
I 547 48 
! 53447 
\ 522 46 


10 

11 
12 
13 

, 14 


4 
5 
5 
6 
6 


4 

5 
5 
5 
6 


2 
3 
3 
3 
3 


2 
2 
3 
3 
3 


2 

2 
2 
3 
3 


373 , 
385 
398 
411 
424 
437 , 
450 
463 
476 
489 


863 2 
888 2 
9132 
939 2 

964 2 


J 137 
5 112 
5 087 
s 061 
5 036 


490 

503 ; 

515| 

527 
540 


I 510 45 
! 49744 
J 485 43 
I 473 42 
] 460 41 


15 

16 
17 
18 
19 


6 
7 

7 
8 
8 


6 
7 
7 

8 
8 


4 
4 
4 

4 
4 


3 
3 
4 
4 

4 


3 

3 
3 


3 563 
t 550 

J 537 
, 524 

3 511 


989 2 
9901 5 2 
0*02 
0652 
0902 


5 Oil 
G 00985 
* 960 
5 935 
5 910 


552 
564 
577 
589 
601 J 


, 44840 
: 43639 
J 423 38 
I 411 37 
j 39936 


20 

21 
22 
23 
24 


9 
9 
10 
10 

10 


8 
9 
9 
10 
10 


5 
5 

5 
5 
6 


4 
5 
5 
5 
5 




25 

26 
27 
28 
29 
30 
31 
32 
33 
34 


502 
515 
5281 
540 ; 

553} 


., 498 
, 485 
J 472 
; 460 
, 447 


116* 
1412 
1662 
1912 
2172 
99242 2 
267 2 
293 2 
318 2 
3432 

368 2 
3942 
419 2 
4442 

469 2 


b 884 
* 859 
5 834 
5 809 
6 783 


614, 
626 
639 
651 
663 
14676 , 
688 l 
701 
713 
726 


, 38635 
f 37434 
^ 361 33 
i 349 32 
2 337 31 


25 

26 
27 
28 
29 


11 

11 

12 
12 

13 


10 

11 

11 

12 
12 


6 
6 

6 

7 
7 


5 
6 
6 
6 

6 


5 
5 
6 
6 


84566 , 
579 
592 
605 
618 


, 15434 
, 421 
, 408 
f 395 
i 382 


J 00758 

r > 733 
o 707 
r > 682 
5 657 


85324 30 
: 31229 
! 299 28 
, 287 27 
f 274 26 


30 

31 
32 
33 
34 


13 

13 
14 

14 
15 


12 
13 

13 
14 
14 


7 

7 
7 
8 
8 


6 
7 
7 

7 
7 


6 

6 
6 

7 

7 


35 

36 
37 

38 
39 


630 
643 
656 
669 
682 


, 370 
I 357 
i 344 
I 331 
2 318 


632 
6 (506 
5 581 
5 556 
5 531 


738 
750 
763 
775 
788: 
"800, 
813 
825 1 
838! 
850 
863 
875 
888 
900 
913 


I 262 25 
2 25024 
* 237 23 
; 225 22 
2 2122 


35 

36 
37 
38 
39 


15 

16 

16 

16 
17 


15 
15 

15 
16 

16 


8 
8 
9 
9 
9 


8 
8 
8 
8 
8 


7 

7 
7 
8 
8 


40 

41 
42 
43 
44 
45 
46 
47 
48 
49 


694 
707 
720 
733 
745 


, 306 
, 293 
, 280 
I 267 
2 255 


495 2 
5202 
5452 
5702 
596 2 
621 2 
6462 
6722 
6972 
7222 


6 505 
5 480 
455 
* 430 
6 404 


. 20020 
, 187 9 
? 175 8 
^ 162 7 
, 150 6 


40 

41 
42 
43 
44 


17 
18 
18 
19 
19 


17 
17 
18 
18 

18 


9 
10 
10 
10 

10 


9 
9 
9 

9 
10 


8 
8 
8 
9 
9 


758, 
771 
784 
796 
809 ] 
"822 
835 ] 
847 
860 
873 


, 242 
? 22* 
o 21( 

8? 


d 379 
5 354 
6 328 
5 303 

5 278 


o 137 15 

2 125 4 
, 112 3 
2 100 2 
i 087 


45 

46 
47 
48 
49 


20 

20 

20 
21 

21 


19 
19 
20 
20 

20 


10 

11 
11 

11 
11 


10 
10 

10 
10 

1] 


9 

9 
9 
10 
10 


>< 
5 
52 
53 
54 


IS 

is 

; 127 


747 i 
7732 
798 2 
823 2 
848 2 


" 253 
6 227 
5 202 
5 177 

5 152 


926 
938 ] 
951 
963 ' 
976 ] 


, 074 
2 062 
I 049 
2 037 
I 024 


50 

51 
52 
53 
54 


22 
22 
23 
23 

23 


21 

21 
22 
22 

22 


12 
12 

12 
12 
13 


11 
11 

11 
11 

12 


10 

10 
10 

11 
11 


55 

56 
57 
58 
59 


885 
898 
911 
923 
936 


- IS 
~ 

L ;j 064 


874^ 
899, 
924 
049 
975: 


5 126 

? 101 
* 076 

A 5 

; 025 


988 
15001 
014 
026 
039 


, 012 
? 84999 
! 986 
I 974 

2 961 


55 

56 
57 
58 
59 


24 

24 
25 
25 

26 


23 

23 
24 
24 

25 


13 
13 

13 
14 
14 


12 

12 
12 
13 
13 


11 

11 
11 

12 
12 


60 


84949 


1505 


00000 


' 00000 


15051 


84949 


60 


26 


25 


14 


13 


12 


9. 

Jcos 


d 10. 

i Zsec 


10. 

I cot 


d 10. 

L' I tan 


10. 

I CSC 


d 9. , 

1 / sin 




26 


25 

Prop* 


14 

Ktional 


13 

Parts 


12 



134 C 



45 C